This course develops the basic language of semiclassical analysis, where one studies oscillatory phenomena with a small parameter $h$ that measures scale. The central goal is to understand how differential operators, oscillatory integrals, and wave propagation behave when one tracks both size and localization simultaneously in phase space. The course begins with the semiclassical viewpoint and model oscillations, then introduces symbol classes and quantization on Euclidean space as the framework for turning functions on phase space into operators.
From there, the course builds the semiclassical pseudodifferential calculus: composition, adjoints, remainders, mapping properties, and the associated semiclassical Sobolev spaces. Ellipticity and parametrices then provide the basic inversion theory, while semiclassical wavefront sets describe where a family of functions or distributions fails to be smooth at a given scale. Positivity results and sharp Garding estimates supply the key lower bounds used in analysis, and the final chapters extend the calculus to manifolds and assemble these tools into a working microlocal toolkit for localizing and solving semiclassical problems in geometric settings.
# Introduction
This opening chapter fixes the viewpoint of the course. Semiclassical analysis studies families of functions, distributions, and operators depending on a small positive parameter $h$, and asks for estimates that remain meaningful as $h \to 0$. The course is built around one guiding principle: oscillation at wavelength $h$ should be measured in phase space, where position $x$ and frequency $\xi$ are treated together. Later chapters develop the symbol and pseudodifferential machinery; this chapter explains the analytic setting and the questions that machinery is designed to answer.
The prerequisites are real and functional analysis, distribution theory, Fourier analysis, Sobolev spaces, elliptic PDE, and smooth manifolds at the level of charts and partitions of unity. The course will use these subjects as working tools rather than redevelop them. In particular, the reader should be comfortable passing between functions and distributions, estimating Fourier transforms, and reading local coordinate formulae on a manifold.
## Why a Semiclassical Parameter Appears
The first question is why a new parameter is worth introducing at all. Many analytic problems involve a family of functions whose oscillation becomes faster and faster, or a family of operators whose coefficients depend on a small scale. Writing that scale as $h$ lets us separate two issues: the ordinary analytic object at each fixed $h$, and the uniform behaviour of the whole family as $h \to 0$.
[definition: Semiclassical Family]
A semiclassical family in a topological vector space $X$ is a family $(u_h)_{0<h\le h_0}$ with $u_h \in X$ for each $h \in (0,h_0]$.
[/definition]
The point of the definition is not the indexing itself, but the estimates imposed on the family. A statement such as $\|u_h\|_X \le C$ means that the constant $C$ is independent of $h$ for all sufficiently small $h$, while a statement such as $\|u_h\|_X = O(h^N)$ describes a rate of decay as the wavelength scale shrinks.
[example: Plane Wave Family]
Let $a \in C_c^\infty(\mathbb R^n)$ and fix a nonzero vector $\xi_0 \in \mathbb R^n$. For $0<h\le h_0$, set
\begin{align*}
u_h(x)=e^{ix\cdot \xi_0/h}a(x).
\end{align*}
Since $e^{ix\cdot \xi_0/h}$ never vanishes, the physical support of $u_h$ is contained in $\operatorname{supp}a$; in fact $u_h(x)=0$ whenever $a(x)=0$.
To see the oscillatory scale, differentiate in the $x_j$ direction. By the product rule,
\begin{align*}
\partial_{x_j}u_h(x)=\partial_{x_j}\!\left(e^{ix\cdot\xi_0/h}\right)a(x)+e^{ix\cdot\xi_0/h}\partial_{x_j}a(x).
\end{align*}
Since
\begin{align*}
x\cdot\xi_0=\sum_{k=1}^n x_k(\xi_0)_k,
\end{align*}
we have
\begin{align*}
\partial_{x_j}\!\left(\frac{i}{h}x\cdot\xi_0\right)=\frac{i}{h}(\xi_0)_j.
\end{align*}
The one-variable chain rule applied to the exponential therefore gives
\begin{align*}
\partial_{x_j}\!\left(e^{ix\cdot\xi_0/h}\right)=\frac{i(\xi_0)_j}{h}e^{ix\cdot\xi_0/h}.
\end{align*}
Substituting this into the product-rule identity yields
\begin{align*}
\partial_{x_j}u_h(x)=h^{-1}i(\xi_0)_j e^{ix\cdot\xi_0/h}a(x)+e^{ix\cdot\xi_0/h}\partial_{x_j}a(x).
\end{align*}
Thus the derivative contains a term of size $h^{-1}$ multiplying the original oscillatory factor, while differentiating the amplitude only contributes the fixed smooth function $\partial_{x_j}a$. This is why ordinary derivatives see wavelength-$h$ oscillation as large, and why semiclassical operators rescale derivatives by powers of $h$.
[/example]
This example shows that ordinary differential operators see wavelength-$h$ oscillation as large. To build operators whose leading behaviour remains finite as $h\to0$, we package derivatives with the same scale as the oscillation; this motivates the basic semiclassical differential operators used throughout the course.
[definition: Semiclassical Differential Operator]
A semiclassical differential operator of order at most $m$ on an open set $U \subset \mathbb R^n$ is a family of linear maps
\begin{align*}
P_h:C^\infty(U)\to C^\infty(U)
\end{align*}
of the form
\begin{align*}
P_hu=\sum_{|\alpha|\le m} a_\alpha(x;h)(hD_x)^\alpha u,
\end{align*}
where $a_\alpha(\cdot;h) \in C^\infty(U)$ for $0<h\le h_0$, $D_{x_j}=(1/i)\partial_{x_j}$, and $(hD_x)^\alpha=\prod_{j=1}^n(hD_{x_j})^{\alpha_j}$.
[/definition]
After this rescaling, the highest-order part of the operator becomes a function on phase space rather than only a differential expression. This is the first appearance of the principal symbol, which later becomes the organising object for ellipticity, parametrices, and propagation.
## Position, Frequency, and Phase Space
The next problem is that neither physical-space localization nor Fourier-space localization alone describes a wave packet. A function may be supported near one point but contain many frequencies, or it may have a narrow band of frequencies but be spread over all space. Semiclassical analysis therefore works on phase space $T^*\mathbb R^n$, whose points are pairs $(x,\xi)$.
[definition: Semiclassical Phase Space]
For an open set $U\subset \mathbb R^n$, the semiclassical phase space over $U$ is the cotangent bundle $T^*U$, identified in coordinates with $U\times \mathbb R^n$. A point of $T^*U$ is written $(x,\xi)$, where $x$ is the position variable and $\xi$ is the semiclassical frequency variable.
[/definition]
This notation is chosen so that the oscillatory factor $e^{ix\cdot\xi_0/h}$ has bounded semiclassical frequency $\xi_0$, not ordinary Fourier frequency $\xi_0/h$. The course will use symbols $a(x,\xi;h)$ as functions on phase space and quantize them into operators acting on $u_h(x)$.
[example: Gaussian Wave Packet]
Fix $x_0,\xi_0\in\mathbb R^n$ and set
\begin{align*}
u_h(x)=h^{-n/4}e^{-|x-x_0|^2/(2h)}e^{i(x-x_0)\cdot\xi_0/h}.
\end{align*}
Since $\left|e^{i(x-x_0)\cdot\xi_0/h}\right|=1$, the density $|u_h(x)|^2$ is
\begin{align*}
|u_h(x)|^2=h^{-n/2}e^{-|x-x_0|^2/h}.
\end{align*}
With the change of variables $y=(x-x_0)/h^{1/2}$, so that $x=x_0+h^{1/2}y$ and $dx=h^{n/2}\,dy$, the total $L^2$ mass is
\begin{align*}
\int_{\mathbb R^n}|u_h(x)|^2\,dx=\int_{\mathbb R^n}e^{-|y|^2}\,dy=\pi^{n/2}.
\end{align*}
For $R>0$, the mass outside the ball $|x-x_0|\le Rh^{1/2}$ is
\begin{align*}
\int_{|x-x_0|>Rh^{1/2}}|u_h(x)|^2\,dx=\int_{|y|>R}e^{-|y|^2}\,dy.
\end{align*}
The right-hand side is independent of $h$ and tends to $0$ as $R\to\infty$, so the physical mass is concentrated at distance comparable to $h^{1/2}$ from $x_0$.
The semiclassical frequency is measured by $hD_{x_j}=(h/i)\partial_{x_j}$. Differentiating the Gaussian factor gives
\begin{align*}
\partial_{x_j}e^{-|x-x_0|^2/(2h)}=-\frac{x_j-(x_0)_j}{h}e^{-|x-x_0|^2/(2h)}.
\end{align*}
Differentiating the oscillatory factor gives
\begin{align*}
\partial_{x_j}e^{i(x-x_0)\cdot\xi_0/h}=\frac{i(\xi_0)_j}{h}e^{i(x-x_0)\cdot\xi_0/h}.
\end{align*}
Using the product rule,
\begin{align*}
hD_{x_j}u_h=(\xi_0)_j u_h+i(x_j-(x_0)_j)u_h.
\end{align*}
Therefore
\begin{align*}
(hD_{x_j}-(\xi_0)_j)u_h=i(x_j-(x_0)_j)u_h.
\end{align*}
Its squared $L^2$ size is
\begin{align*}
\|(hD_{x_j}-(\xi_0)_j)u_h\|_{L^2}^2=\int_{\mathbb R^n}(x_j-(x_0)_j)^2h^{-n/2}e^{-|x-x_0|^2/h}\,dx.
\end{align*}
Using the same change of variables,
\begin{align*}
\|(hD_{x_j}-(\xi_0)_j)u_h\|_{L^2}^2=h\int_{\mathbb R^n}y_j^2e^{-|y|^2}\,dy.
\end{align*}
Thus the frequency error has size $O(h^{1/2})$ in $L^2$, while the position error $x-x_0$ has the same $h^{1/2}$ scale. The packet is therefore localized near $(x_0,\xi_0)$ in phase space at the uncertainty scale.
[/example]
The uncertainty scale is the reason that phase-space localization is never pointwise in both variables. Instead of asking for the value of a function at $(x,\xi)$, we test it using symbols and operators whose resolution is compatible with $h$.
[remark: Physical and Semiclassical Frequencies]
The ordinary Fourier variable for $e^{ix\cdot\xi_0/h}$ is $\xi_0/h$. The semiclassical frequency variable is $\xi_0$. The course consistently uses $\xi$ for the semiclassical frequency variable.
[/remark]
## Uniform Estimates as the Main Language
The central analytic question is not whether an estimate holds for each individual $h$, but whether it holds with constants independent of $h$. This distinction drives the notation for boundedness, smoothing, ellipticity, and remainders throughout the course.
[definition: Semiclassical Boundedness]
Let $X$ be a normed space. A semiclassical family $(u_h)_{0<h\le h_0}$ in $X$ is bounded in $X$ if there exist $C>0$ and $h_1\in(0,h_0]$ such that
\begin{align*}
\|u_h\|_X\le C
\end{align*}
for all $0<h\le h_1$.
[/definition]
This definition turns asymptotic analysis into ordinary norm estimates with quantified constants. Later, the same convention will apply to operator families $A_h:X\to Y$, where boundedness means $\|A_hu\|_Y\le C\|u\|_X$ uniformly for small $h$.
Some errors are smaller than every finite power of $h$, and those errors should be treated as negligible by the calculus. To express this without choosing a particular exponent, we introduce the standard rapid-decay notation in the semiclassical parameter.
[definition: Rapid Decay in the Semiclassical Parameter]
Let $X$ be a normed space. A semiclassical family $(u_h)_{0<h\le h_0}$ is $O(h^\infty)$ in $X$ if for every $N\in\mathbb N$ there exist $C_N>0$ and $h_N\in(0,h_0]$ such that
\begin{align*}
\|u_h\|_X\le C_N h^N
\end{align*}
for all $0<h\le h_N$.
[/definition]
The notation $O(h^\infty)$ is stronger than any fixed power estimate. It is the natural replacement for negligible error in semiclassical calculus, because composition formulae and parametrices typically produce remainders that are smaller than $h^N$ for every prescribed $N$.
[example: Exponentially Small Family]
For $X=\mathbb R$, set $u_h=e^{-1/h}$. We show that this family is $O(h^\infty)$ by proving a power estimate for each fixed $N\in\mathbb N$. Put $t=1/h$, so that $h^{-N}e^{-1/h}=t^Ne^{-t}$. Since the exponential series has nonnegative terms for $t>0$,
\begin{align*}
e^t=\sum_{k=0}^\infty \frac{t^k}{k!}\ge \frac{t^{N+1}}{(N+1)!}.
\end{align*}
Taking reciprocals gives
\begin{align*}
e^{-t}\le (N+1)!t^{-(N+1)}.
\end{align*}
Multiplying by $t^N$ yields
\begin{align*}
t^Ne^{-t}\le (N+1)!t^{-1}.
\end{align*}
Returning to $h=1/t$, this becomes
\begin{align*}
h^{-N}e^{-1/h}\le (N+1)!h.
\end{align*}
If $0<h\le 1$, then $(N+1)!h\le (N+1)!$, so
\begin{align*}
e^{-1/h}\le (N+1)!h^N.
\end{align*}
Thus for each $N$ the definition holds with $C_N=(N+1)!$ and $h_N=1$, and therefore $e^{-1/h}=O(h^\infty)$ in $\mathbb R$.
[/example]
This example shows why $O(h^\infty)$ is a scale-dependent notion rather than a statement of literal vanishing. The next basic fact explains why such errors can be ignored after applying uniformly controlled operators: the rapid decay survives the operation.
[quotetheorem:7285]
[citeproof:7285]
The theorem records a basic bookkeeping principle used repeatedly later. Once an error is $O(h^\infty)$, every operator estimate with uniform constants preserves its negligible character.
The word "uniform" is essential. If $X=Y=\mathbb R$, $u_h=e^{-1/h}$, and $A_h$ is multiplication by $e^{1/(2h)}$, then $u_h=O(h^\infty)$ but $A_hu_h=e^{-1/(2h)}$ is still rapidly decaying only because the growth was subexponential relative to $e^{1/h}$; multiplication by $e^{1/h}$ would turn the same negligible family into the constant family $1$. Thus the theorem does not say that arbitrary $h$-dependent operations preserve remainders. It says that the calculus may discard $O(h^\infty)$ errors precisely at stages where the operator bounds have already been proved uniformly in $h$.
This distinction becomes important in the construction of parametrices. The symbolic algebra produces approximate inverses up to controlled remainders, and the analytic estimates decide whether those remainders stay negligible after further operators are applied.
## What the First Part of the Course Builds
The remaining question for this introduction is how the parts of the course fit together. The first lectures establish the scaling rules: the semiclassical Fourier transform, the operators $hD_{x_j}$, and integration by parts for phases with no stationary points. These tools explain why oscillation can disappear from an integral when the phase gradient stays away from zero.
The next block develops symbol classes. A symbol is a controlled function $a(x,\xi;h)$ on phase space, with estimates for derivatives in $x$ and $\xi$. These estimates allow us to distinguish order, regularity, and dependence on $h$ before turning the symbol into an operator.
Quantization is the bridge from phase space back to analysis on functions. Given a symbol $a$, a quantization procedure produces an operator $\operatorname{Op}_h(a)$. Different quantizations may differ by lower-order terms, but the calculus is designed so that principal symbolic information is stable.
Pseudodifferential calculus then supplies the algebra. It explains how to compose operators, take adjoints, construct parametrices for elliptic symbols, and track remainders in powers of $h$. This is where the course turns phase-space intuition into reliable estimates.
The final lectures introduce microlocal regularity. Instead of asking whether $u_h$ is smooth everywhere, we ask where in phase space it fails to be negligible. This leads to the semiclassical wavefront set, which records the positions and frequencies at which a family has persistent singular or oscillatory behaviour.
[definition: Semiclassical Wavefront Set]
Let $(u_h)$ be a semiclassical family of distributions on an open set $U\subset\mathbb R^n$. The semiclassical wavefront set $\operatorname{WF}_h(u)$ is the subset of $T^*U$ consisting of phase-space points at which $u_h$ is not microlocally $O(h^\infty)$.
[/definition]
The full definition requires pseudodifferential cutoffs, so this block is only a preview. Its role here is to name the destination: the course develops symbols and quantization so that statements about concentration, oscillation, and singularity can be made locally in phase space.
[example: Expected Wavefront of a Plane Wave]
For $u_h(x)=e^{ix\cdot\xi_0/h}a(x)$ with $a\in C_c^\infty(\mathbb R^n)$, the physical part is immediate: if $x\notin\operatorname{supp}a$, then $a(x)=0$, hence
\begin{align*}
u_h(x)=e^{ix\cdot\xi_0/h}a(x)=0.
\end{align*}
Thus no phase-space point lying over $x\notin\operatorname{supp}a$ can contribute to the wavefront set.
Now let $\chi\in C_c^\infty(\mathbb R^n_\xi)$ be supported away from $\xi_0$, say $|\xi-\xi_0|\ge \delta>0$ on $\operatorname{supp}\chi$. The frequency cutoff $\chi(hD)$ acts by
\begin{align*}
\chi(hD)u_h(x)=(2\pi h)^{-n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\xi/h}\chi(\xi)e^{iy\cdot\xi_0/h}a(y)\,dy\,d\xi.
\end{align*}
Combining the exponentials gives
\begin{align*}
e^{i(x-y)\cdot\xi/h}e^{iy\cdot\xi_0/h}=e^{ix\cdot\xi/h}e^{iy\cdot(\xi_0-\xi)/h}.
\end{align*}
For fixed $\xi$ in $\operatorname{supp}\chi$, set $v=\xi_0-\xi$. Since $|v|\ge\delta$, the operator
\begin{align*}
L_\xi=\frac{h}{i|v|^2}v\cdot\nabla_y
\end{align*}
satisfies
\begin{align*}
L_\xi e^{iy\cdot v/h}=e^{iy\cdot v/h},
\end{align*}
because $v\cdot\nabla_y e^{iy\cdot v/h}=i|v|^2h^{-1}e^{iy\cdot v/h}$. Integrating by parts once transfers $L_\xi$ from the exponential to $a$ and gives a factor $h$:
\begin{align*}
\int_{\mathbb R^n}e^{iy\cdot v/h}a(y)\,dy=\int_{\mathbb R^n}e^{iy\cdot v/h}L_\xi^*a(y)\,dy.
\end{align*}
Since the coefficients of $L_\xi^*$ are bounded by constants depending on $\delta$ and each application differentiates only the fixed compactly supported function $a$, repeating this $N$ times gives
\begin{align*}
\left|\int_{\mathbb R^n}e^{iy\cdot(\xi_0-\xi)/h}a(y)\,dy\right|\le C_N h^N
\end{align*}
uniformly for $\xi\in\operatorname{supp}\chi$. Therefore $\chi(hD)u_h=O(h^\infty)$ after any frequency cutoff supported away from $\xi_0$. The expected semiclassical wavefront set is consequently contained in $\operatorname{supp}a\times\{\xi_0\}$: the function vanishes away from $\operatorname{supp}a$, and all semiclassical frequencies away from $\xi_0$ are killed up to rapidly decaying remainders.
[/example]
This example is the template for the course. Semiclassical analysis begins with ordinary oscillatory functions, translates their behaviour into phase-space language, and then builds an operator calculus robust enough to prove estimates uniformly as $h\to0$.
The opening example has now given the course its central template: oscillations are read through phase space, and operators are built to control them uniformly as $h\to 0$. The next chapter makes that philosophy precise by fixing the semiclassical scale and showing how even simple model oscillations encode the geometry of localization and differentiation.
# 1. Semiclassical Scaling and Model Oscillation
Semiclassical analysis studies families of functions and operators depending on a small parameter $h \in (0,h_0]$. The parameter is not a perturbation parameter in the naive sense: it changes the scale at which oscillation, differentiation, and localization are measured. This opening chapter fixes the basic normalizations used throughout the course and builds a catalogue of model families whose behaviour will later be encoded by symbols and quantization.
The guiding principle is that an $h$-oscillatory factor such as $e^{i x \cdot \xi_0/h}$ has ordinary derivatives of size $h^{-1}$, but has semiclassical derivatives of size $1$. Thus the right variables are not only position $x$, but phase-space variables $(x,\xi)$, where $\xi$ records frequency at the scale set by $h$.
## Uniform Estimates for Parameter-Dependent Families
The first question is how to say that a statement is stable as $h \to 0$. Many objects in the course depend on $h$, and the point is usually not to compute their exact limit, but to keep bounds whose constants do not deteriorate as the wavelength shrinks.
[definition: Semiclassical Family]
Let $X$ be a set or function space and let $h_0>0$. A semiclassical family in $X$ is a collection $(u_h)_{0<h\le h_0}$ with $u_h \in X$ for each $h \in (0,h_0]$.
[/definition]
This definition identifies the parameter range, but it does not yet say what kind of control survives as $h$ changes. The next step is to put a norm on the ambient space so that stability can be expressed by a constant independent of $h$.
[definition: Uniform Bound]
Let $(X,\|\cdot\|_X)$ be a normed space. A semiclassical family $(u_h)_{0<h\le h_0}$ in $X$ is uniformly bounded in $X$ if there exists $C>0$ such that
\begin{align*}
\|u_h\|_X \le C
\end{align*}
for every $h \in (0,h_0]$.
[/definition]
Uniform boundedness is the basic language for estimates, but many remainders are not merely bounded: they come with a definite power of $h$. The next notation records the size of such a family relative to a chosen norm.
[definition: Semiclassical Order]
Let $(u_h)_{0<h\le h_0}$ be a family in a normed space $X$, and let $m \in \mathbb R$. We write $u_h = O_X(h^m)$ if there exist $C>0$ and $h_1 \in (0,h_0]$ such that
\begin{align*}
\|u_h\|_X \le C h^m
\end{align*}
for every $h \in (0,h_1]$.
[/definition]
Positive powers of $h$ represent small remainders, while negative powers record controlled blow-up. The following model shows why the norm matters: the same family may be harmless in $L^2$ and large after ordinary differentiation.
[example: Oscillatory Plane Wave With Fixed Amplitude]
Let $a \in C_c^\infty(\mathbb R^n)$ and $\xi_0 \in \mathbb R^n$, and define
\begin{align*}
u_h(x)=e^{i x\cdot \xi_0/h}a(x).
\end{align*}
Since $|e^{i x\cdot \xi_0/h}|=1$ for every $x$, we have
\begin{align*}
\|u_h\|_{L^2}^2=\int_{\mathbb R^n}|e^{i x\cdot \xi_0/h}a(x)|^2\,d\mathcal L^n(x)=\int_{\mathbb R^n}|a(x)|^2\,d\mathcal L^n(x)=\|a\|_{L^2}^2.
\end{align*}
Thus $\|u_h\|_{L^2}=\|a\|_{L^2}$ for all $h$, so $(u_h)$ is uniformly bounded in $L^2(\mathbb R^n)$.
For an ordinary derivative in the $x_j$ direction, the product rule gives
\begin{align*}
\partial_{x_j}u_h(x)=\partial_{x_j}\left(e^{i x\cdot \xi_0/h}\right)a(x)+e^{i x\cdot \xi_0/h}\partial_{x_j}a(x).
\end{align*}
Since $\partial_{x_j}(x\cdot \xi_0)=\xi_{0,j}$, the chain rule gives
\begin{align*}
\partial_{x_j}\left(e^{i x\cdot \xi_0/h}\right)=\frac{i\xi_{0,j}}{h}e^{i x\cdot \xi_0/h}.
\end{align*}
Therefore
\begin{align*}
\partial_{x_j}u_h(x)=e^{i x\cdot \xi_0/h}\left(\frac{i\xi_{0,j}}{h}a(x)+\partial_{x_j}a(x)\right).
\end{align*}
Taking the $L^2$ norm and again using $|e^{i x\cdot \xi_0/h}|=1$,
\begin{align*}
\|\partial_{x_j}u_h\|_{L^2}=\left\|\frac{i\xi_{0,j}}{h}a+\partial_{x_j}a\right\|_{L^2}.
\end{align*}
By the reverse triangle inequality,
\begin{align*}
\|\partial_{x_j}u_h\|_{L^2}\ge \frac{|\xi_{0,j}|}{h}\|a\|_{L^2}-\|\partial_{x_j}a\|_{L^2}.
\end{align*}
Hence, if $a\ne0$ and some $\xi_{0,j}\ne0$, then $\|\partial_{x_j}u_h\|_{L^2}\to\infty$ as $h\to0$. The family is therefore uniformly bounded in $L^2$, but not uniformly bounded in $H^1$ when $\xi_0\ne0$ and $a$ is nonzero; if $\xi_0=0$, then $u_h=a$ is independent of $h$ and is uniformly bounded in every ordinary Sobolev norm.
[/example]
This example is the first indication that ordinary Sobolev norms are not adapted to oscillation at wavelength $h$. We want derivatives that regard $e^{i x\cdot \xi_0/h}$ as having frequency $\xi_0$, not frequency $\xi_0/h$.
## The Semiclassical Fourier Transform
The next question is how to rescale Fourier analysis so that the variable dual to $x$ is the semiclassical frequency $\xi$. The usual Fourier transform sees oscillations at physical frequency $\xi/h$; the semiclassical transform renames $h$ times that frequency as the principal variable. Throughout this chapter, $\mathcal L^n$ denotes Lebesgue measure on $\mathbb R^n$.
[definition: Semiclassical Fourier Transform]
For $h>0$, the semiclassical Fourier transform is the linear map $\mathcal F_h:\mathcal S(\mathbb R^n)\to \mathcal S(\mathbb R^n)$ defined by
\begin{align*}
\mathcal F_h u(\xi)
:= (2\pi h)^{-n/2}\int_{\mathbb R^n} e^{-i x\cdot \xi/h}u(x)\,d\mathcal L^n(x).
\end{align*}
Its inverse on $\mathcal S(\mathbb R^n)$ is the linear map $\mathcal F_h^{-1}:\mathcal S(\mathbb R^n)\to \mathcal S(\mathbb R^n)$ defined by
\begin{align*}
\mathcal F_h^{-1}v(x)
= (2\pi h)^{-n/2}\int_{\mathbb R^n} e^{i x\cdot \xi/h}v(\xi)\,d\mathcal L^n(\xi).
\end{align*}
[/definition]
We also use the ordinary symmetric Fourier transform $\mathcal F:\mathcal S(\mathbb R^n)\to\mathcal S(\mathbb R^n)$, with $\widehat f=\mathcal F f$, defined by
\begin{align*}
\mathcal F f(\eta)=\widehat f(\eta):=(2\pi)^{-n/2}\int_{\mathbb R^n}e^{-ix\cdot\eta}f(x)\,d\mathcal L^n(x).
\end{align*}
With this convention, $\mathcal F_h$ is the ordinary transform evaluated at the rescaled frequency, with the normalization adjusted to preserve the $L^2$ norm.
The rescaling by $h$ would be much less useful if it distorted the Hilbert space scale: energy estimates, orthogonality arguments, and density extensions all depend on measuring the input and output in the same $L^2$ norm. The possible obstruction is the factor $(2\pi h)^{-n/2}$ in the integral, which could change total mass unless it is exactly balanced by the frequency rescaling. The formal statement records that the semiclassical transform is genuinely unitary on $L^2$, not just a convenient oscillatory integral on Schwartz functions.
[quotetheorem:247]
[citeproof:247]
Plancherel says that replacing the usual transform by $\mathcal F_h$ does not change the Hilbert space geometry. The initial Schwartz-space hypothesis is used to justify the integral formula directly; for a general $L^2$ function, the integral need not define an ordinary function pointwise, so the transform is understood by density and completion. A concrete obstruction is the characteristic function $u=\mathbb{1}_{[0,1]^n}$: it belongs to $L^2(\mathbb R^n)$, but it is neither smooth nor rapidly decreasing, and the integral formula must be interpreted through the $L^2$ extension rather than as a Schwartz-to-Schwartz calculation. The $L^2$ hypothesis in the extension cannot be dropped: the constant function $u(x)=1$ is not in $L^2(\mathbb R^n)$, and there is no finite $L^2$ norm for a unitary identity to preserve. The theorem does not say that $\mathcal F_hu$ is pointwise controlled or localized in frequency, only that the total $L^2$ mass is preserved. Its role is to make oscillatory families appear at bounded frequency, which is the starting point for the phase-space bookkeeping used in microlocal analysis and in the quantum-mechanical interpretation of $h$ as Planck-scale resolution.
[example: Pure Frequency Concentration]
Let $a\in\mathcal S(\mathbb R^n)$, fix $\xi_0\in\mathbb R^n$, and set $u_h(x)=e^{i x\cdot \xi_0/h}a(x)$. We compute its semiclassical Fourier transform from the definition:
\begin{align*}
\mathcal F_hu_h(\xi)=(2\pi h)^{-n/2}\int_{\mathbb R^n}e^{-ix\cdot\xi/h}e^{ix\cdot\xi_0/h}a(x)\,d\mathcal L^n(x).
\end{align*}
The two exponential factors combine by adding exponents:
\begin{align*}
e^{-ix\cdot\xi/h}e^{ix\cdot\xi_0/h}=e^{-ix\cdot(\xi-\xi_0)/h}.
\end{align*}
Substituting this into the integral gives
\begin{align*}
\mathcal F_hu_h(\xi)=(2\pi h)^{-n/2}\int_{\mathbb R^n}e^{-ix\cdot(\xi-\xi_0)/h}a(x)\,d\mathcal L^n(x).
\end{align*}
By the definition of $\mathcal F_h a$ evaluated at $\xi-\xi_0$,
\begin{align*}
\mathcal F_hu_h(\xi)=\mathcal F_ha(\xi-\xi_0).
\end{align*}
Thus multiplication by $e^{ix\cdot\xi_0/h}$ shifts the semiclassical Fourier transform by $\xi_0$: wherever $\mathcal F_ha$ is concentrated near $0$, the transformed family $\mathcal F_hu_h$ is concentrated near $\xi_0$.
[/example]
The formula explains why $\xi$ is the right dual variable for the rest of the course. A phase $x\cdot \xi_0/h$ records the covector $\xi_0$, not the larger ordinary frequency $\xi_0/h$.
## Semiclassical Derivatives and Frequency Scaling
The transform becomes most useful when paired with a derivative convention. The problem is that $\partial_{x_j}$ turns a model oscillation into a factor of size $h^{-1}$, while the associated phase-space frequency should remain order $1$.
[definition: Semiclassical Differential Operator]
For $h>0$ and $1\le j\le n$, the $j$th semiclassical differential operator is the linear map $hD_{x_j}:\mathcal S(\mathbb R^n)\to \mathcal S(\mathbb R^n)$ defined by
\begin{align*}
hD_{x_j}u=\frac{h}{i}\partial_{x_j}u.
\end{align*}
For a multi-index $\alpha=(\alpha_1,\dots,\alpha_n)$, define the linear map $(hD_x)^\alpha:\mathcal S(\mathbb R^n)\to \mathcal S(\mathbb R^n)$ by
\begin{align*}
(hD_x)^\alpha u=(hD_{x_1})^{\alpha_1}\cdots (hD_{x_n})^{\alpha_n}u.
\end{align*}
[/definition]
The factor $h$ compensates for the wavelength, and the factor $1/i$ matches the Fourier convention. The next identity confirms that this operator is exactly the physical realization of multiplication by the semiclassical momentum variable.
[quotetheorem:249]
[citeproof:249]
The theorem is the computational bridge between differentiation and phase-space localization. The Schwartz hypothesis keeps the integration by parts free of boundary terms and decay issues. A concrete failure occurs on the half-line if the same integration-by-parts argument is attempted for $u(x)=e^{-x}\mathbb{1}_{[0,\infty)}(x)$: the boundary at $x=0$ contributes a term, so differentiating under the Fourier integral is not the same as multiplying by $\xi$ without an additional boundary contribution. On $\mathbb R$, the function $u(x)=(1+|x|)^{-1/2}$ is locally integrable but not in $L^1$ or $\mathcal S$, so the defining oscillatory integral is not an absolutely convergent Schwartz integral. The identity also does not say that ordinary derivatives are uniformly bounded: a plane wave still has $\partial_{x_j}u_h$ of size $h^{-1}$ even though $hD_{x_j}u_h$ is order one. It motivates the semiclassical Sobolev norms that appear later, where powers of $hD_x$ replace powers of ordinary derivatives and make propagation estimates compatible with fixed phase-space frequency.
[example: Semiclassical Derivatives of a Plane Wave]
For $u_h(x)=e^{i x\cdot \xi_0/h}a(x)$ with $a\in C_c^\infty(\mathbb R^n)$, the definition $hD_{x_j}=(h/i)\partial_{x_j}$ gives
\begin{align*}
hD_{x_j}u_h(x)=\frac{h}{i}\partial_{x_j}\left(e^{i x\cdot \xi_0/h}a(x)\right).
\end{align*}
By the product rule,
\begin{align*}
\partial_{x_j}\left(e^{i x\cdot \xi_0/h}a(x)\right)=\partial_{x_j}\left(e^{i x\cdot \xi_0/h}\right)a(x)+e^{i x\cdot \xi_0/h}\partial_{x_j}a(x).
\end{align*}
Since $\partial_{x_j}(x\cdot\xi_0)=\xi_{0,j}$, the chain rule gives
\begin{align*}
\partial_{x_j}\left(e^{i x\cdot \xi_0/h}\right)=\frac{i\xi_{0,j}}{h}e^{i x\cdot \xi_0/h}.
\end{align*}
Substituting this derivative into the previous identity,
\begin{align*}
hD_{x_j}u_h(x)=\frac{h}{i}\left(\frac{i\xi_{0,j}}{h}e^{i x\cdot \xi_0/h}a(x)+e^{i x\cdot \xi_0/h}\partial_{x_j}a(x)\right).
\end{align*}
Multiplying the two terms by $h/i$ gives
\begin{align*}
hD_{x_j}u_h(x)=\xi_{0,j}e^{i x\cdot \xi_0/h}a(x)+e^{i x\cdot \xi_0/h}hD_{x_j}a(x).
\end{align*}
Because $|e^{i x\cdot \xi_0/h}|=1$,
\begin{align*}
\|hD_{x_j}u_h\|_{L^2}\le |\xi_{0,j}|\|a\|_{L^2}+\|hD_{x_j}a\|_{L^2}.
\end{align*}
Using $hD_{x_j}a=(h/i)\partial_{x_j}a$,
\begin{align*}
\|hD_{x_j}a\|_{L^2}=h\|\partial_{x_j}a\|_{L^2}\le h_0\|\partial_{x_j}a\|_{L^2}.
\end{align*}
Thus $hD_{x_j}u_h$ is uniformly bounded in $L^2$ for $0<h\le h_0$. Moreover,
\begin{align*}
\left\|hD_{x_j}u_h-\xi_{0,j}e^{i x\cdot \xi_0/h}a\right\|_{L^2}=h\|\partial_{x_j}a\|_{L^2}.
\end{align*}
So the semiclassical derivative acts on this plane wave as multiplication by $\xi_{0,j}$ up to an $L^2$ error of order $h$, which is the model calculation behind identifying $hD_x$ with the momentum variable $\xi$.
[/example]
## Uniform Schwartz Control
After rescaling derivatives, we need a way to track families whose decay and differentiability remain controlled at every order. The Schwartz class is the natural testing ground because both position weights and frequency weights can be estimated without boundary issues.
[definition: Uniform Schwartz Family]
A semiclassical family $(u_h)_{0<h\le h_0}$ in $\mathcal S(\mathbb R^n)$ is uniformly Schwartz if for every pair of multi-indices $\alpha,\beta$ there exists $C_{\alpha\beta}>0$ such that
\begin{align*}
\sup_{0<h\le h_0}\sup_{x\in\mathbb R^n}|x^\alpha D^\beta u_h(x)|\le C_{\alpha\beta}.
\end{align*}
[/definition]
This is an ordinary Schwartz condition imposed uniformly over the parameter interval. The next estimate relates this control to semiclassical derivatives and to the scaled Fourier-side seminorms that will later reappear in symbol estimates.
[quotetheorem:7286]
[citeproof:7286]
The estimate warns against confusing ordinary Schwartz bounds in $\xi$ with scaled Schwartz bounds in $\xi/h$. Uniformity in the Schwartz seminorms is essential: for example, $u_h=h^{-1}a$ with fixed nonzero $a\in\mathcal S(\mathbb R^n)$ is Schwartz for each $h$ but has seminorms blowing up as $h\to0$, already at the $C^0$ seminorm. The first assertion would therefore fail for $\alpha=\beta=0$ if uniformity were omitted. The second assertion is stated for a fixed profile, and that hypothesis is separate from pointwise membership in $\mathcal S$. For a precise failing $h$-dependent profile, take $a_h(x)=h^{-1}a(x)$ with fixed nonzero $a\in\mathcal S(\mathbb R^n)$. Then
\begin{align*}
\mathcal F_h a_h(\xi)=h^{-1}\mathcal F_h a(\xi),
\end{align*}
so the displayed Fourier-side seminorm in the theorem is multiplied by $h^{-1}$ and is not uniformly bounded. The fixed-profile hypothesis matters because the identity $\mathcal F_h a(\xi)=h^{-n/2}\mathcal F a(\xi/h)$ reduces the estimate to ordinary Schwartz seminorms of one function. If the profile is allowed to vary with $h$, derivatives, weights, or amplitudes may introduce additional powers of $h$ that are not controlled by this argument. A fixed physical profile has a semiclassical transform that narrows as $h\to0$, so its ordinary pointwise height may grow even though its $L^2$ norm stays fixed, a distinction that later becomes part of symbol estimates and uncertainty-principle scaling.
[example: Rescaled Bump and Localization Scales]
Let $\chi\in C_c^\infty(\mathbb R^n)$, fix $x_0\in\mathbb R^n$, and define
\begin{align*}
u_h(x)=h^{-n/2}\chi\left(\frac{x-x_0}{h}\right).
\end{align*}
First compute the $L^2$ norm. Since
\begin{align*}
|u_h(x)|^2=h^{-n}\left|\chi\left(\frac{x-x_0}{h}\right)\right|^2,
\end{align*}
the change of variables $y=(x-x_0)/h$, so $x=x_0+hy$ and $d\mathcal L^n(x)=h^n\,d\mathcal L^n(y)$, gives
\begin{align*}
\|u_h\|_{L^2}^2=\int_{\mathbb R^n}h^{-n}\left|\chi\left(\frac{x-x_0}{h}\right)\right|^2\,d\mathcal L^n(x)=\int_{\mathbb R^n}|\chi(y)|^2\,d\mathcal L^n(y)=\|\chi\|_{L^2}^2.
\end{align*}
Thus $\|u_h\|_{L^2}=\|\chi\|_{L^2}$ for every $h>0$. If $\operatorname{supp}\chi\subset B(0,R)$, then $u_h(x)\ne0$ only when $(x-x_0)/h\in B(0,R)$, equivalently $|x-x_0|<hR$, so $u_h$ is localized in a ball of radius proportional to $h$ around $x_0$.
Now compute the semiclassical Fourier transform from the definition:
\begin{align*}
\mathcal F_hu_h(\xi)=(2\pi h)^{-n/2}\int_{\mathbb R^n}e^{-ix\cdot\xi/h}h^{-n/2}\chi\left(\frac{x-x_0}{h}\right)\,d\mathcal L^n(x).
\end{align*}
Using the same substitution $x=x_0+hy$,
\begin{align*}
\mathcal F_hu_h(\xi)=(2\pi h)^{-n/2}h^{-n/2}h^n\int_{\mathbb R^n}e^{-i(x_0+hy)\cdot\xi/h}\chi(y)\,d\mathcal L^n(y).
\end{align*}
The prefactor satisfies
\begin{align*}
(2\pi h)^{-n/2}h^{-n/2}h^n=(2\pi)^{-n/2}.
\end{align*}
Also,
\begin{align*}
e^{-i(x_0+hy)\cdot\xi/h}=e^{-ix_0\cdot\xi/h}e^{-iy\cdot\xi}.
\end{align*}
Therefore
\begin{align*}
\mathcal F_hu_h(\xi)=e^{-ix_0\cdot\xi/h}(2\pi)^{-n/2}\int_{\mathbb R^n}e^{-iy\cdot\xi}\chi(y)\,d\mathcal L^n(y)=e^{-ix_0\cdot\xi/h}\widehat{\chi}(\xi).
\end{align*}
Since $|e^{-ix_0\cdot\xi/h}|=1$, the frequency density is $|\mathcal F_hu_h(\xi)|=|\widehat{\chi}(\xi)|$, independent of $h$. Thus shrinking the physical support to scale $h$ produces an order-one spread in the semiclassical frequency variable $\xi$.
[/example]
This example separates two notions that will recur throughout microlocal analysis. A family may be localized in physical space, in frequency space, or in both; the correct bookkeeping uses the pair $(x,\xi)$.
## Nonstationary Semiclassical Phase
Oscillatory integrals enter semiclassical analysis whenever a function is represented by a phase and an amplitude. The first estimate asks what happens when the phase has no critical point on the support of the amplitude: in that case, oscillation forces cancellation faster than any fixed power of $h$.
[quotetheorem:7287]
[citeproof:7287]
The lemma is the basic mechanism behind the phrase "microlocally absent": if a phase has no stationary point compatible with a region of phase space, repeated integration by parts makes the contribution negligible to all algebraic orders in $h$. The lower bound on $|\nabla\phi|$ is indispensable; when $\phi(x)=x^2$ near $0$, a stationary point remains and the corresponding oscillatory integral has a leading size of order $h^{1/2}$ rather than $O(h^N)$ for every $N$. Compact support of the amplitude is also part of the mechanism, since it removes boundary terms and keeps the coefficients of the integration-by-parts operator controlled on a fixed compact set. The result does not describe the leading term in the stationary case; that is the role of stationary phase, which later connects these estimates to PDE propagation and microlocal parametrices.
[example: Linear Phase Away From Zero Frequency]
Let $a\in C_c^\infty(\mathbb R^n)$, let $\xi_0\ne0$, and set
\begin{align*}
I(h)=\int_{\mathbb R^n}e^{i x\cdot \xi_0/h}a(x)\,d\mathcal L^n(x).
\end{align*}
Use the phase $\phi(x)=x\cdot\xi_0$. For each coordinate $x_j$,
\begin{align*}
\partial_{x_j}\phi(x)=\partial_{x_j}\left(\sum_{k=1}^n x_k\xi_{0,k}\right)=\xi_{0,j}.
\end{align*}
Hence
\begin{align*}
\nabla\phi(x)=(\xi_{0,1},\dots,\xi_{0,n})=\xi_0.
\end{align*}
Therefore
\begin{align*}
|\nabla\phi(x)|=|\xi_0|>0
\end{align*}
for every $x\in\mathbb R^n$. Since $a$ has compact support, $K=\operatorname{supp}a$ is compact, and the lower bound $|\nabla\phi|\ge |\xi_0|$ holds on $K$.
By the *Integration by Parts Lemma for Nonstationary Semiclassical Phase*, for every $N\in\mathbb N$ there is a constant $C_N>0$ such that
\begin{align*}
|I(h)|\le C_Nh^N
\end{align*}
for all sufficiently small $h>0$. Thus
\begin{align*}
I(h)=O(h^N)
\end{align*}
for every $N$.
With the ordinary symmetric Fourier transform convention,
\begin{align*}
\widehat a(\eta)=(2\pi)^{-n/2}\int_{\mathbb R^n}e^{-ix\cdot\eta}a(x)\,d\mathcal L^n(x).
\end{align*}
Taking $\eta=-\xi_0/h$ gives
\begin{align*}
\widehat a(-\xi_0/h)=(2\pi)^{-n/2}\int_{\mathbb R^n}e^{ix\cdot\xi_0/h}a(x)\,d\mathcal L^n(x).
\end{align*}
Multiplying by $(2\pi)^{n/2}$,
\begin{align*}
I(h)=(2\pi)^{n/2}\widehat a(-\xi_0/h).
\end{align*}
So the same estimate says that the Fourier transform of the smooth compactly supported amplitude is being sampled at the ordinary frequency $-\xi_0/h$, whose magnitude tends to infinity as $h\to0$.
[/example]
This estimate is the first place where smoothness and oscillation trade against each other. Later, the same idea will appear in symbol calculus when remainders are controlled by integrating away variables whose phases have no critical points.
## Wave Packets and Phase-Space Concentration
The final question of the chapter is how to model a family localized near both a point $x_0$ in space and a covector $\xi_0$ in frequency. Pure plane waves have precise frequency but no spatial localization; small bumps have spatial localization but spread in frequency. Wave packets balance these effects at the scale compatible with the uncertainty principle.
[definition: Semiclassical Wave Packet]
Let $h_0>0$, let $x_0,\xi_0\in\mathbb R^n$, and let $g\in\mathcal S(\mathbb R^n)$. The associated semiclassical wave packet is the family $(u_h)_{0<h\le h_0}$ with $u_h\in\mathcal S(\mathbb R^n)$ and $u_h:\mathbb R^n\to\mathbb C$ given by
\begin{align*}
u_h:x\mapsto h^{-n/4}g\left(\frac{x-x_0}{h^{1/2}}\right)e^{i(x-x_0)\cdot\xi_0/h}.
\end{align*}
[/definition]
The normalization preserves the $L^2$ size, the factor $h^{1/2}$ gives spatial width $h^{1/2}$, and the oscillatory exponential centers the semiclassical frequency at $\xi_0$. The Gaussian case lets us compute these claims explicitly and then formulate the phase-space center as a theorem.
[example: Gaussian Coherent State]
Take $g(y)=\pi^{-n/4}e^{-|y|^2/2}$ and define
\begin{align*}
u_h(x)=h^{-n/4}\pi^{-n/4}e^{-|x-x_0|^2/(2h)}e^{i(x-x_0)\cdot\xi_0/h}.
\end{align*}
Since the oscillatory factor has modulus $1$,
\begin{align*}
|u_h(x)|^2=h^{-n/2}\pi^{-n/2}e^{-|x-x_0|^2/h}.
\end{align*}
With $y=(x-x_0)/h^{1/2}$, so $x=x_0+h^{1/2}y$ and $d\mathcal L^n(x)=h^{n/2}\,d\mathcal L^n(y)$, we get
\begin{align*}
\|u_h\|_{L^2}^2=\pi^{-n/2}\int_{\mathbb R^n}e^{-|y|^2}\,d\mathcal L^n(y).
\end{align*}
The product Gaussian integral gives $\int_{\mathbb R^n}e^{-|y|^2}\,d\mathcal L^n(y)=\pi^{n/2}$, hence
\begin{align*}
\|u_h\|_{L^2}^2=1.
\end{align*}
Moreover, for every $R>0$,
\begin{align*}
\int_{|x-x_0|>Rh^{1/2}}|u_h(x)|^2\,d\mathcal L^n(x)=\pi^{-n/2}\int_{|y|>R}e^{-|y|^2}\,d\mathcal L^n(y).
\end{align*}
The right-hand side is independent of $h$ and tends to $0$ as $R\to\infty$, so the spatial mass is concentrated at distance of order $h^{1/2}$ from $x_0$.
Now compute the semiclassical Fourier transform:
\begin{align*}
\mathcal F_hu_h(\xi)=(2\pi h)^{-n/2}h^{-n/4}\pi^{-n/4}\int_{\mathbb R^n}e^{-ix\cdot\xi/h}e^{-|x-x_0|^2/(2h)}e^{i(x-x_0)\cdot\xi_0/h}\,d\mathcal L^n(x).
\end{align*}
Using the same substitution $x=x_0+h^{1/2}y$,
\begin{align*}
e^{-ix\cdot\xi/h}e^{i(x-x_0)\cdot\xi_0/h}=e^{-ix_0\cdot\xi/h}e^{-iy\cdot(\xi-\xi_0)/h^{1/2}}.
\end{align*}
The prefactor becomes
\begin{align*}
(2\pi h)^{-n/2}h^{-n/4}\pi^{-n/4}h^{n/2}=(2\pi)^{-n/2}h^{-n/4}\pi^{-n/4}.
\end{align*}
Therefore
\begin{align*}
\mathcal F_hu_h(\xi)=e^{-ix_0\cdot\xi/h}(2\pi)^{-n/2}h^{-n/4}\pi^{-n/4}\int_{\mathbb R^n}e^{-|y|^2/2}e^{-iy\cdot(\xi-\xi_0)/h^{1/2}}\,d\mathcal L^n(y).
\end{align*}
The standard Gaussian Fourier integral
\begin{align*}
(2\pi)^{-n/2}\int_{\mathbb R^n}e^{-|y|^2/2}e^{-iy\cdot\eta}\,d\mathcal L^n(y)=e^{-|\eta|^2/2}
\end{align*}
with $\eta=(\xi-\xi_0)/h^{1/2}$ gives
\begin{align*}
\mathcal F_hu_h(\xi)=h^{-n/4}\pi^{-n/4}e^{-ix_0\cdot\xi/h}e^{-|\xi-\xi_0|^2/(2h)}.
\end{align*}
Thus
\begin{align*}
|\mathcal F_hu_h(\xi)|^2=h^{-n/2}\pi^{-n/2}e^{-|\xi-\xi_0|^2/h}.
\end{align*}
With $\zeta=(\xi-\xi_0)/h^{1/2}$,
\begin{align*}
\int_{|\xi-\xi_0|>Rh^{1/2}}|\mathcal F_hu_h(\xi)|^2\,d\mathcal L^n(\xi)=\pi^{-n/2}\int_{|\zeta|>R}e^{-|\zeta|^2}\,d\mathcal L^n(\zeta).
\end{align*}
This again tends to $0$ as $R\to\infty$, independently of $h$, so the semiclassical frequency mass is concentrated at distance of order $h^{1/2}$ from $\xi_0$. The packet has unit $L^2$ mass, spatial center $x_0$, semiclassical frequency center $\xi_0$, and balanced width $h^{1/2}$ in both variables.
[/example]
Gaussian packets give the simplest picture of phase-space concentration, but the picture is still qualitative until one specifies how the center and width are being measured. Pointwise size is not the right invariant measurement, because Gaussians have tails and are never supported in a compact phase-space box.
The remaining question is whether the parameters $x_0$ and $\xi_0$ in the formula are merely suggestive labels, or whether they can be recovered canonically from the packet itself. The useful test is by first and second moments of the spatial density and of the semiclassical Fourier density; these moments distinguish the chosen center from the unavoidable $h^{1/2}$ uncertainty scale and give a quantitative version of phase-space localization.
[quotetheorem:7288]
[citeproof:7288]
This theorem gives a concrete interpretation of phase-space localization before any pseudodifferential operators have been introduced. Gaussian normalization is not cosmetic: if $v_h=2u_h$, then
\begin{align*}
\int_{\mathbb R^n}x|v_h(x)|^2\,d\mathcal L^n(x)=4x_0,
\qquad
\int_{\mathbb R^n}\xi|\mathcal F_hv_h(\xi)|^2\,d\mathcal L^n(\xi)=4\xi_0,
\end{align*}
so the unnormalized moment integrals measure total mass as well as centre. The Gaussian profile also matters for the exact centre statement: if a normalized profile is changed to $h^{-n/4}g((x-x_0)/h^{1/2})e^{i(x-x_0)\cdot\xi_0/h}$ with $g\in\mathcal S(\mathbb R^n)$ and $\int y|g(y)|^2\,d\mathcal L^n(y)\ne0$, then the spatial expectation becomes $x_0+h^{1/2}\int y|g(y)|^2\,d\mathcal L^n(y)$ rather than exactly $x_0$. The theorem also does not say that the packet is supported in a small phase-space box; Gaussians have tails, so the statement is about concentration through moments rather than compact support. The later calculus will generalize this measurement: symbols will test where a family lives in $(x,\xi)$, operators will act by manipulating that localization, and the width $h^{1/2}$ will reappear as the balanced scale in the uncertainty principle.
[remark: Three Model Families]
Plane waves $e^{i x\cdot\xi_0/h}a(x)$ model frequency concentration near $\xi_0$ with spatial profile $a$. Rescaled bumps $h^{-n/2}\chi((x-x_0)/h)$ model strong physical localization with order-one semiclassical frequency spread. Coherent states combine localization near $x_0$ and $\xi_0$ at the balanced scale $h^{1/2}$.
[/remark]
These models form the vocabulary for the next chapter. Symbol classes will give uniform estimates for functions of $(x,\xi)$, and quantization will turn those symbols into operators acting on the oscillatory families introduced here.
With the basic oscillatory models in place, we can now classify the amplitudes that act on them. The next chapter turns the phase-space intuition from semiclassical scaling into symbol classes, where uniform $h$-dependent estimates become the language for quantization.
# 2. Semiclassical Symbol Classes
This chapter turns the phase-space intuition from semiclassical scaling into a calculus of amplitudes. In Chapter 1, oscillations such as $e^{ix\cdot \xi_0/h}$ showed that the relevant variables are position $x$, frequency $\xi$, and the small parameter $h$. The task now is to decide which $h$-dependent functions $a(x,\xi;h)$ are regular enough, have controlled growth enough, and admit enough asymptotic structure to serve as symbols for quantization.
The chapter has three connected aims. First, we introduce order functions and symbol classes on $T^*\mathbb R^n$, including the mildly $h$-singular classes $S^m_\delta$. Second, we isolate the leading terms of a symbol through principal and subprincipal symbols, and prove the asymptotic summation theorem that makes formal expansions meaningful. Third, we record the compactly supported and residual variants that are used later when localizing to small regions of phase space.
## Measuring Growth in Phase Space
The first problem is that symbols are not required to be bounded on $T^*\mathbb R^n$. Differential operators have polynomial growth in $\xi$, potentials may grow or decay in $x$, and parametrices often require weights that combine both variables. We therefore need a flexible way to measure allowed growth before imposing derivative estimates.
[definition: Japanese Bracket]
The Japanese bracket is the function $\langle\cdot\rangle:\mathbb R^n\to[1,\infty)$ defined by
\begin{align*}
\xi \longmapsto \langle \xi \rangle := (1 + |\xi|^2)^{1/2}.
\end{align*}
[/definition]
The Japanese bracket replaces $|\xi|$ by a smooth positive function that behaves like $1$ near $\xi=0$ and like $|\xi|$ for large $|\xi|$. It is the standard weight for symbols whose growth is polynomial only in the fiber variable.
[example: Polynomial Fiber Weight]
For $m\in\mathbb R$, set $a(x,\xi;h)=\langle\xi\rangle^m=(1+|\xi|^2)^{m/2}$. It is independent of $x$ and $h$, so every $x$-derivative vanishes. We show that each $\xi$-derivative has one lower power of $\langle\xi\rangle$ per derivative.
For one derivative, the chain rule gives
\begin{align*}
\partial_{\xi_j}\langle\xi\rangle^m=m\xi_j(1+|\xi|^2)^{m/2-1}
\end{align*}
and since $|\xi_j|\le |\xi|\le \langle\xi\rangle$,
\begin{align*}
|\partial_{\xi_j}\langle\xi\rangle^m|\le |m|\langle\xi\rangle\langle\xi\rangle^{m-2}=|m|\langle\xi\rangle^{m-1}.
\end{align*}
For higher derivatives, repeated use of the product rule and chain rule gives a finite sum of terms of the form
\begin{align*}
c_{\beta,k,\gamma}\xi^\gamma(1+|\xi|^2)^{m/2-k}
\end{align*}
where $|\gamma|\le 2k-|\beta|$. Each such term satisfies
\begin{align*}
|\xi^\gamma(1+|\xi|^2)^{m/2-k}|\le \langle\xi\rangle^{|\gamma|}\langle\xi\rangle^{m-2k}\le \langle\xi\rangle^{m-|\beta|}.
\end{align*}
Because only finitely many terms occur for a fixed $\beta$, there is a constant $C_\beta>0$ such that
\begin{align*}
|\partial_\xi^\beta\langle\xi\rangle^m|\le C_\beta\langle\xi\rangle^{m-|\beta|}.
\end{align*}
Thus $\langle\xi\rangle^m$ is the model fiber weight of order $m$: differentiating in $\xi$ improves the fiber growth by the number of derivatives.
[/example]
The preceding example also raises the problem of choosing weights beyond the single model $\langle\xi\rangle^m$. We need a definition that permits position-dependent and anisotropic growth while still preventing the weight from changing faster than polynomially across phase space.
[definition: Order Function]
An order function on $T^*\mathbb R^n$ is a function $m:T^*\mathbb R^n \to (0,\infty)$ such that there exist constants $C>0$ and $N\ge 0$ with
\begin{align*}
m(x,\xi) \le C\langle (x,\xi)-(y,\eta)\rangle^N m(y,\eta)
\end{align*}
for all $(x,\xi),(y,\eta) \in T^*\mathbb R^n$.
[/definition]
This condition says that the weight may grow, but only at a polynomial rate relative to the phase-space distance. The basic examples are $m(x,\xi)=\langle \xi\rangle^r$, $m(x,\xi)=\langle x\rangle^s\langle \xi\rangle^r$, and the constant weight $m=1$.
[remark: Why Order Functions Are Used]
The order-function condition is weaker than requiring a specific formula such as $\langle\xi\rangle^m$. It allows the calculus to treat local weights, anisotropic weights, and products of position and frequency weights with the same notation. Later, elliptic estimates will often be phrased by saying that a symbol is bounded below by an order function.
[/remark]
Order functions solve the problem of allowed growth, but they do not yet prevent an amplitude from oscillating faster and faster as $h\to0$ or from developing uncontrolled derivatives in phase space. Such behavior would break integration by parts, composition estimates, and the separation between a symbol's size and the oscillation carried by the phase. The symbol class therefore imposes uniform derivative bounds relative to the chosen order function, so that $a(x,\xi;h)$ remains a controlled amplitude throughout the semiclassical limit.
[definition: Semiclassical Symbol Class]
Let $m$ be an order function on $T^*\mathbb R^n$. The class $S(m)$ consists of all functions
\begin{align*}
a:T^*\mathbb R^n \times (0,h_0] \longrightarrow \mathbb C
\end{align*}
with $a \in C^\infty(T^*\mathbb R^n \times (0,h_0];\mathbb C)$ such that for every pair of multi-indices $\alpha,\beta$ there is $C_{\alpha\beta}>0$ with
\begin{align*}
|\partial_x^\alpha \partial_\xi^\beta a(x,\xi;h)| \le C_{\alpha\beta} m(x,\xi)
\end{align*}
for all $(x,\xi)\in T^*\mathbb R^n$ and $0<h\le h_0$.
[/definition]
The constants may depend on the derivatives being taken, but not on $x$, $\xi$, or $h$. This is what lets symbolic estimates survive the limit $h\to0$. We next need the standard shorthand for the most common choice of order function, where growth is measured only by powers of the fiber weight.
[definition: Standard Symbol Class Of Order M]
For $m\in\mathbb R$, define
\begin{align*}
S^m(T^*\mathbb R^n) := S(\langle \xi\rangle^m).
\end{align*}
[/definition]
Thus $S^0$ contains uniformly bounded symbols with uniformly bounded derivatives, while $S^2$ contains the principal symbols of many second-order differential operators. The definition does not force derivatives in $\xi$ to lower the order; it only requires the weaker estimate by the original weight. Some theorems use the sharper standard class in which fiber derivatives improve decay. We record that notation now so later hypotheses can distinguish the two conventions.
[definition: Standard $(1,0)$ Symbol Class]
For $m\in\mathbb R$, the class $S^m_{1,0}(T^*\mathbb R^n)$ consists of all $a\in C^\infty(T^*\mathbb R^n\times(0,h_0])$ such that for every pair of multi-indices $\alpha,\beta$ there is a constant $C_{\alpha\beta}>0$ with
\begin{align*}
|\partial_x^\alpha\partial_\xi^\beta a(x,\xi;h)|\le C_{\alpha\beta}\langle\xi\rangle^{m-|\beta|}
\end{align*}
for all $(x,\xi)\in T^*\mathbb R^n$ and $0<h\le h_0$.
[/definition]
The subscript $(1,0)$ means that each $\xi$-derivative lowers the allowed fiber order by one, while $x$-derivatives do not change the order. This is the familiar Hormander convention adapted to the semiclassical parameter. It is stronger than the broad $S^m$ notation above, and it is the class in which high-frequency parametrices and conversion formulas have their cleanest form. In practice, many important examples satisfy these stronger estimates, and the next examples and stability theorem show why the simpler order-function convention is already enough for much of the calculus.
[example: Schrodinger Symbol]
Let $V\in C^\infty(\mathbb R^n;\mathbb R)$, assume $V$ is bounded, and assume $\partial_x^\alpha V$ is bounded for every multi-index $\alpha$ with $|\alpha|\ge 1$. For fixed $E\in\mathbb R$, define
\begin{align*}
p(x,\xi)=|\xi|^2+V(x)-E.
\end{align*}
We verify from the definition of $S^2(T^*\mathbb R^n)$ that every derivative is bounded by a constant times $\langle\xi\rangle^2$, uniformly in $(x,\xi)$.
For the zeroth derivative,
\begin{align*}
|p(x,\xi)|\le |\xi|^2+|V(x)|+|E|.
\end{align*}
Since $|\xi|^2\le \langle\xi\rangle^2$ and $1\le \langle\xi\rangle^2$,
\begin{align*}
|p(x,\xi)|\le \bigl(1+\|V\|_{L^\infty}+|E|\bigr)\langle\xi\rangle^2.
\end{align*}
For first fiber derivatives,
\begin{align*}
\partial_{\xi_j}p(x,\xi)=2\xi_j.
\end{align*}
Thus
\begin{align*}
|\partial_{\xi_j}p(x,\xi)|=2|\xi_j|\le 2|\xi|\le 2\langle\xi\rangle\le 2\langle\xi\rangle^2.
\end{align*}
For second fiber derivatives,
\begin{align*}
\partial_{\xi_j}\partial_{\xi_k}p(x,\xi)=2\delta_{jk}.
\end{align*}
Hence
\begin{align*}
|\partial_{\xi_j}\partial_{\xi_k}p(x,\xi)|\le 2\le 2\langle\xi\rangle^2.
\end{align*}
Every fiber derivative of order at least three is zero because $|\xi|^2$ is quadratic in $\xi$ and $V(x)-E$ is independent of $\xi$.
For base derivatives, if $|\alpha|\ge 1$ and $\beta=0$, then
\begin{align*}
\partial_x^\alpha p(x,\xi)=\partial_x^\alpha V(x).
\end{align*}
By the boundedness assumption on derivatives of $V$,
\begin{align*}
|\partial_x^\alpha p(x,\xi)|\le \|\partial_x^\alpha V\|_{L^\infty}\le \|\partial_x^\alpha V\|_{L^\infty}\langle\xi\rangle^2.
\end{align*}
If both $|\alpha|\ge 1$ and $|\beta|\ge 1$, then
\begin{align*}
\partial_x^\alpha\partial_\xi^\beta p(x,\xi)=0,
\end{align*}
because the $\xi$-dependent part $|\xi|^2$ has no $x$-dependence and the $x$-dependent part $V(x)-E$ has no $\xi$-dependence.
Therefore, for every pair of multi-indices $\alpha,\beta$, there is a constant $C_{\alpha\beta}>0$ such that
\begin{align*}
|\partial_x^\alpha\partial_\xi^\beta p(x,\xi)|\le C_{\alpha\beta}\langle\xi\rangle^2.
\end{align*}
Since $p$ is independent of $h$, the same estimates are uniform in $0<h\le h_0$, so $p\in S^2(T^*\mathbb R^n)$.
[/example]
The example shows a single differential expression belongs to a symbol class, but the calculus needs closure under the operations that build new expressions from old ones. We therefore need the following stability theorem for differentiation and multiplication; it is the algebraic backbone used before quantization enters.
[quotetheorem:7289]
[citeproof:7289]
The hypotheses in this theorem are doing real work. If $a$ is merely pointwise bounded by $m_1$ but its derivatives are not controlled by the same order function, the first conclusion can fail after a single differentiation; for instance, $a(x,\xi;h)=\sin(e^{x_1})$ is bounded, but $\partial_{x_1}a=e^{x_1}\cos(e^{x_1})$ is not controlled by the constant order function. The multiplication statement also requires using the product order function $m_1m_2$: multiplying $\langle\xi\rangle^r$ by $\langle\xi\rangle^s$ generally raises the growth order to $r+s$, so there is no reason for the product to remain in either original class.
The theorem is therefore a closure statement, not a smoothing or order-lowering statement. It preserves the amount of growth allowed by the inputs and records how much growth has been added. This is why exact constants in symbol estimates are rarely named: the calculus is controlled by families of seminorms, and each operation only needs finitely many of them. The same bookkeeping will reappear when the symbols are quantized, where composition estimates are built from repeated differentiations and products.
## Mildly Singular Semiclassical Classes
The standard class $S(m)$ treats differentiation uniformly in $h$. Many semiclassical constructions require symbols that become sharper as $h\to0$, such as cutoffs to phase-space balls whose radius depends on $h$. The parameter $\delta$ measures how much loss in powers of $h$ is allowed per derivative.
[definition: Delta Symbol Class]
Let $0\le \delta < 1/2$, and let $m$ be an order function on $T^*\mathbb R^n$. The class $S_\delta(m)$ consists of all functions
\begin{align*}
a:T^*\mathbb R^n\times(0,h_0]\longrightarrow \mathbb C
\end{align*}
with $a\in C^\infty(T^*\mathbb R^n\times(0,h_0];\mathbb C)$ such that for every pair of multi-indices $\alpha,\beta$ there is $C_{\alpha\beta}>0$ with
\begin{align*}
|\partial_x^\alpha\partial_\xi^\beta a(x,\xi;h)| \le C_{\alpha\beta} h^{-\delta(|\alpha|+|\beta|)}m(x,\xi)
\end{align*}
for all $(x,\xi)\in T^*\mathbb R^n$ and $0<h\le h_0$.
[/definition]
The restriction $0\le\delta<1/2$ is the range in which the usual semiclassical symbolic expansions retain positive powers of $h$ after composing operators. The endpoint and beyond require different bookkeeping, so this course keeps to the stable range.
[definition: Delta Symbol Class Of Order M]
For $m\in\mathbb R$ and $0\le\delta<1/2$, define
\begin{align*}
S^m_\delta(T^*\mathbb R^n):=S_\delta(\langle\xi\rangle^m).
\end{align*}
[/definition]
These classes contain the standard classes, since $S^m\subset S^m_\delta$ for every allowed $\delta$. The inclusion is useful but not reversible: $h$-dependent localization can create derivative losses absent in $S^m$.
[example: Phase-Space Cutoff At Scale H Delta]
Fix $\chi\in C_c^\infty(\mathbb R^{2n})$ and $(x_0,\xi_0)\in T^*\mathbb R^n$, and write
\begin{align*}
y(x,\xi;h)=\left(\frac{x-x_0}{h^\delta},\frac{\xi-\xi_0}{h^\delta}\right).
\end{align*}
We show that the symbol
\begin{align*}
a(x,\xi;h)=\chi(y(x,\xi;h))
\end{align*}
belongs to $S_\delta(1)$, meaning that every mixed derivative is bounded by a constant times $h^{-\delta(|\alpha|+|\beta|)}$.
For one $x$-derivative, the chain rule gives
\begin{align*}
\partial_{x_j}a(x,\xi;h)=h^{-\delta}(\partial_{y_j}\chi)(y(x,\xi;h)).
\end{align*}
For one $\xi$-derivative, where the $\xi$ variables occupy the last $n$ coordinates of $\mathbb R^{2n}$, the chain rule gives
\begin{align*}
\partial_{\xi_j}a(x,\xi;h)=h^{-\delta}(\partial_{y_{n+j}}\chi)(y(x,\xi;h)).
\end{align*}
Iterating this computation, each $x$-derivative differentiates one of the first $n$ variables of $\chi$ and contributes a factor $h^{-\delta}$, while each $\xi$-derivative differentiates one of the last $n$ variables of $\chi$ and contributes the same factor. Thus, for multi-indices $\alpha,\beta\in\mathbb N^n$,
\begin{align*}
\partial_x^\alpha\partial_\xi^\beta a(x,\xi;h)=h^{-\delta(|\alpha|+|\beta|)}(\partial_y^{(\alpha,\beta)}\chi)(y(x,\xi;h)).
\end{align*}
Since $\partial_y^{(\alpha,\beta)}\chi$ is smooth and compactly supported, there is a finite constant
\begin{align*}
C_{\alpha\beta}:=\sup_{y\in\mathbb R^{2n}}|\partial_y^{(\alpha,\beta)}\chi(y)|.
\end{align*}
Therefore
\begin{align*}
|\partial_x^\alpha\partial_\xi^\beta a(x,\xi;h)|\le C_{\alpha\beta}h^{-\delta(|\alpha|+|\beta|)}.
\end{align*}
This is exactly the defining estimate for $S_\delta(1)$.
The support also shrinks in phase space: $a(x,\xi;h)$ can be nonzero only when $y(x,\xi;h)\in\operatorname{supp}\chi$, so the cutoff localizes to an $h^\delta$-scale neighbourhood of $(x_0,\xi_0)$ while remaining controlled in the $S_\delta(1)$ seminorms.
[/example]
The example shows that $S_\delta$ contains the scale-dependent cutoffs needed for microlocalization, but those cutoffs must still support symbolic algebra. We therefore need the following stability result, which carries the derivative loss through differentiation and multiplication.
[quotetheorem:7290]
[citeproof:7290]
The fixed value of $\delta$ is part of the hypothesis. The product proof compares all derivatives against one common loss factor $h^{-\delta(|\alpha|+|\beta|)}$; if two symbols are given only with unrelated losses, the product belongs at best to the class governed by the larger loss, assuming that larger loss is still below $1/2$. For example, a cutoff at scale $h^{\delta_1}$ multiplied by a cutoff at scale $h^{\delta_2}$ differentiates at the sharper of the two scales, so it is naturally controlled with $\max(\delta_1,\delta_2)$ rather than with the smaller parameter.
The theorem also does not say that the full semiclassical calculus remains unchanged for every $\delta$. The algebraic estimates above make sense at the level of differentiating and multiplying functions, but the later operator composition expansion loses powers of $h^{1-2\delta}$; this is why the course keeps $0\le\delta<1/2$. Near the endpoint $\delta=1/2$, the formal remainder no longer gains a positive power of $h$, so symbolic expansions stop improving in the way needed for the standard calculus. Thus $S_\delta$ records controlled shrinking of phase-space support, while the restriction on $\delta$ protects the later asymptotic machinery.
## Classical Symbols And Leading Terms
A symbol class controls size, but it does not yet say which part of a symbol is most important as $h\to0$. The next problem is to give a precise meaning to formal expansions such as $a\sim a_0+h a_1+h^2a_2+\cdots$.
[definition: Classical Semiclassical Symbol]
Let $m$ be an order function. A symbol $a\in S(m)$ is classical, written $a\in S_{\mathrm{cl}}(m)$, if there exist $h$-independent functions
\begin{align*}
a_j:T^*\mathbb R^n\longrightarrow \mathbb C,\qquad a_j\in C^\infty(T^*\mathbb R^n;\mathbb C),\qquad j\ge0,
\end{align*}
satisfying the symbol estimates of $S(m)$ uniformly as constant families in $h$, such that for every $N\ge0$,
\begin{align*}
a-\sum_{j=0}^{N-1} h^j a_j \in h^N S(m).
\end{align*}
[/definition]
The expansion is an asymptotic statement in the topology of symbol seminorms. It does not claim convergence of the series for any fixed $h$; it says that finite truncations approximate the symbol to arbitrarily high powers of $h$. We next need a way to extract the first layer of this expansion without depending on all higher-order choices.
[definition: Principal Symbol]
Let $a\in S(m)$. The principal symbol of $a$ is its equivalence class in the quotient
\begin{align*}
S(m)/hS(m).
\end{align*}
[/definition]
If $a$ is classical with expansion $a\sim a_0+h a_1+h^2a_2+\cdots$, then this quotient class is represented by $a_0$. The quotient viewpoint is important because the representative is determined modulo $hS(m)$ before a particular expansion is chosen. In computations, we usually write the principal symbol as an $h$-independent function, but the invariant statement is the quotient class.
[definition: Subprincipal Symbol]
Let $a\in S_{\mathrm{cl}}(m)$ have an expansion
\begin{align*}
a\sim a_0+h a_1+h^2a_2+\cdots.
\end{align*}
The subprincipal term of the symbol $a$ is the coefficient $a_1$ in this expansion.
[/definition]
The word "term" is used here deliberately. At the symbol level no quantization convention is needed to name the coefficient $a_1$. Later, for operators, the subprincipal symbol may include correction terms depending on the quantization convention; that is a separate operator-level notion.
[example: Classical Expansion With A Potential]
Let $V_0,V_1\in C_b^\infty(\mathbb R^n)$ and
\begin{align*}
p(x,\xi;h)=|\xi|^2+V_0(x)-E+hV_1(x).
\end{align*}
Define
\begin{align*}
p_0(x,\xi)=|\xi|^2+V_0(x)-E,\qquad p_1(x,\xi)=V_1(x),\qquad p_j=0\text{ for }j\ge2.
\end{align*}
We check that these coefficients lie in $S^2(T^*\mathbb R^n)$ and that they give the required classical expansion.
For $p_0$, the zeroth derivative satisfies
\begin{align*}
|p_0(x,\xi)|\le |\xi|^2+\|V_0\|_{L^\infty}+|E|\le \bigl(1+\|V_0\|_{L^\infty}+|E|\bigr)\langle\xi\rangle^2.
\end{align*}
If $|\beta|=1$, then $\partial_\xi^\beta|\xi|^2$ is one of the functions $2\xi_j$, so
\begin{align*}
|\partial_\xi^\beta p_0(x,\xi)|\le 2|\xi|\le 2\langle\xi\rangle\le 2\langle\xi\rangle^2.
\end{align*}
If $|\beta|=2$, then $\partial_\xi^\beta|\xi|^2$ is either $2$ or $0$, hence
\begin{align*}
|\partial_\xi^\beta p_0(x,\xi)|\le 2\le 2\langle\xi\rangle^2.
\end{align*}
If $|\beta|\ge3$, then
\begin{align*}
\partial_\xi^\beta p_0(x,\xi)=0.
\end{align*}
For base derivatives, if $|\alpha|\ge1$ and $\beta=0$, then
\begin{align*}
\partial_x^\alpha\partial_\xi^\beta p_0(x,\xi)=\partial_x^\alpha V_0(x),
\end{align*}
so
\begin{align*}
|\partial_x^\alpha p_0(x,\xi)|\le \|\partial_x^\alpha V_0\|_{L^\infty}\le \|\partial_x^\alpha V_0\|_{L^\infty}\langle\xi\rangle^2.
\end{align*}
If $|\alpha|\ge1$ and $|\beta|\ge1$, then
\begin{align*}
\partial_x^\alpha\partial_\xi^\beta p_0(x,\xi)=0,
\end{align*}
because $|\xi|^2$ is independent of $x$ and $V_0(x)-E$ is independent of $\xi$. Thus $p_0\in S^2(T^*\mathbb R^n)$.
For $p_1=V_1(x)$, every $\xi$-derivative with $|\beta|\ge1$ vanishes:
\begin{align*}
\partial_x^\alpha\partial_\xi^\beta p_1(x,\xi)=0.
\end{align*}
When $\beta=0$, boundedness of the derivatives of $V_1$ gives
\begin{align*}
|\partial_x^\alpha p_1(x,\xi)|=|\partial_x^\alpha V_1(x)|\le \|\partial_x^\alpha V_1\|_{L^\infty}\langle\xi\rangle^2.
\end{align*}
Hence $p_1\in S^2(T^*\mathbb R^n)$, and $p_j=0\in S^2(T^*\mathbb R^n)$ for every $j\ge2$.
The expansion is exact at first order:
\begin{align*}
p(x,\xi;h)-p_0(x,\xi)=h p_1(x,\xi).
\end{align*}
After subtracting the $h$ term, the remainder is zero:
\begin{align*}
p(x,\xi;h)-p_0(x,\xi)-h p_1(x,\xi)=0.
\end{align*}
Therefore the $N=1$ remainder lies in $hS^2(T^*\mathbb R^n)$, and for every $N\ge2$ the remainder lies in $h^N S^2(T^*\mathbb R^n)$ because it is the zero symbol. Thus $p\in S^2_{\mathrm{cl}}(T^*\mathbb R^n)$ with principal term $p_0(x,\xi)=|\xi|^2+V_0(x)-E$ and subprincipal term $p_1(x,\xi)=V_1(x)$. The point of the example is that a finite $h$-polynomial is classical, with no hidden higher-order remainder.
[/example]
The principal symbol behaves as the first nonzero layer of a filtered algebra. The following exactness statement explains how vanishing of the principal symbol is the same as divisibility by $h$ inside the symbol class.
[quotetheorem:7291]
[citeproof:7291]
This exactness principle depends on working with the quotient by the precise ideal $hS(m)$. If the denominator were replaced by a smaller class, such as $hS(m')$ where $m'\le m$ and $m/m'$ is unbounded at fiber infinity, the quotient would be too fine. For instance, with $m(x,\xi)=\langle\xi\rangle^r$ and $m'(x,\xi)=\langle\xi\rangle^s$ for $s<r$, the symbol $h\langle\xi\rangle^r$ has zero leading term in the intended order $r$ sense but does not lie in $hS(m')$. If the denominator were made larger, leading information would be lost: modding out by $S(m)$ itself would identify every symbol with zero. The quotient therefore records exactly one power of semiclassical improvement and no more.
The principle is used constantly. When an operator identity is proved modulo $h$, it becomes an identity of principal symbols; when the leading symbol vanishes, the object improves by one power of $h$. It does not identify lower-order behavior in $\xi$ unless that lower order is accompanied by a power of $h$, so principal-symbol exactness is a statement about the semiclassical filtration rather than about decay at fiber infinity.
## Asymptotic Expansions And Summation
Formal expansions are useful only if they are not merely notation for nonexistent sums. A sequence of coefficients $a_0,a_1,a_2,\dots$ may have seminorms that grow so quickly that the raw series $\sum h^j a_j$ diverges for every fixed $h>0$, so ordinary convergence is the wrong requirement. What is needed is an actual symbol whose finite truncation errors have the prescribed powers of $h$ in the symbol topology, with any remaining ambiguity pushed into $h^\infty S(m)$.
[quotetheorem:7292]
[citeproof:7292]
The theorem is a Borel-type result for symbol spaces. It justifies the notation $a\sim\sum_{j\ge0}h^j a_j$: the series is formal, but there is an actual symbol with those prescribed coefficients. The conclusion is only unique modulo $h^\infty S(m)$ because every finite list of coefficients is blind to a residual correction; adding $e^{-1/h}b(x,\xi)$ with $b\in S(m)$ changes the symbol but changes no finite coefficient.
The hypotheses also explain why ordinary convergence is not claimed. The formal series may diverge for every fixed $h>0$ if the seminorms of $a_j$ grow too fast, so the proof inserts cutoffs depending on $j$ and $h$ rather than summing the raw series. Without control of the coefficient sequence in the symbol seminorms, finite truncations would not have remainders in $h^N S(m)$, and the resulting object would not be a symbol with the prescribed asymptotic expansion. Summation therefore turns formal coefficient data into a symbol only after accepting the natural ambiguity by residual terms.
[remark: Meaning Of H Infinity]
The notation $r\in h^\infty S(m)$ means that for every $N\ge0$, the family $h^{-N}r$ belongs to $S(m)$. Such remainders are smaller than every algebraic power of $h$ in all symbol seminorms. They are not generally zero, but they are invisible to every finite-order asymptotic expansion.
[/remark]
The asymptotic summation theorem also makes it possible to define operations on classical symbols coefficient by coefficient. If a construction produces a formal sequence of candidate coefficients, summation turns that sequence into a symbol, and uniqueness modulo $h^\infty$ makes the result independent of irrelevant choices.
## Compact Support And Microlocal Localization
The next problem is local rather than asymptotic. Microlocal arguments rarely use a symbol on all of phase space at once; they insert cutoffs that isolate a small region around a point $(x_0,\xi_0)$ or a conic region near fiber infinity.
[definition: Compactly Supported Symbol]
A symbol $a\in S(m)$ is compactly supported in phase space if there is a compact set $K\subset T^*\mathbb R^n$ such that
\begin{align*}
\operatorname{supp}_{(x,\xi)} a(\cdot,\cdot;h) \subset K
\end{align*}
for every $0<h\le h_0$.
[/definition]
The support condition is uniform in $h$. This uniformity matters because otherwise the family could drift to infinity as $h\to0$ while each individual symbol remained compactly supported.
[example: Localization Near A Phase-Space Point]
Let $\chi_x,\chi_\xi\in C_c^\infty(\mathbb R^n)$ satisfy $\chi_x=1$ near $0$ and $\chi_\xi=1$ near $0$. For fixed $(x_0,\xi_0)\in T^*\mathbb R^n$ and $\varepsilon>0$, define
\begin{align*}
a(x,\xi)=\chi_x\left(\frac{x-x_0}{\varepsilon}\right)\chi_\xi\left(\frac{\xi-\xi_0}{\varepsilon}\right).
\end{align*}
The support is compact because $a(x,\xi)\ne0$ implies $(x-x_0)/\varepsilon\in\operatorname{supp}\chi_x$ and $(\xi-\xi_0)/\varepsilon\in\operatorname{supp}\chi_\xi$, hence
\begin{align*}
\operatorname{supp}a\subset \bigl(x_0+\varepsilon\operatorname{supp}\chi_x\bigr)\times\bigl(\xi_0+\varepsilon\operatorname{supp}\chi_\xi\bigr).
\end{align*}
The set on the right is compact, so $a\in C_c^\infty(T^*\mathbb R^n)$.
For multi-indices $\alpha,\beta$, the chain rule gives
\begin{align*}
\partial_x^\alpha\partial_\xi^\beta a(x,\xi)=\varepsilon^{-|\alpha|-|\beta|}(\partial^\alpha\chi_x)\left(\frac{x-x_0}{\varepsilon}\right)(\partial^\beta\chi_\xi)\left(\frac{\xi-\xi_0}{\varepsilon}\right).
\end{align*}
Since $\partial^\alpha\chi_x$ and $\partial^\beta\chi_\xi$ are smooth and compactly supported, their suprema are finite, so
\begin{align*}
|\partial_x^\alpha\partial_\xi^\beta a(x,\xi)|\le \varepsilon^{-|\alpha|-|\beta|}\|\partial^\alpha\chi_x\|_{L^\infty}\|\partial^\beta\chi_\xi\|_{L^\infty}.
\end{align*}
Because $1$ is the order function for $S(1)$ and $a$ is independent of $h$, these estimates are uniform in $0<h\le h_0$, proving $a\in S(1)$.
Choose radii $r_x,r_\xi>0$ such that $\chi_x(z)=1$ for $|z|<r_x$ and $\chi_\xi(\zeta)=1$ for $|\zeta|<r_\xi$. If $|x-x_0|<\varepsilon r_x$ and $|\xi-\xi_0|<\varepsilon r_\xi$, then
\begin{align*}
a(x,\xi)=1\cdot1=1.
\end{align*}
Thus $a$ is a compactly supported symbol equal to $1$ on a smaller neighbourhood of $(x_0,\xi_0)$, which is exactly the localizer needed to restrict an argument to a prescribed bounded region of phase space.
[/example]
The example handles bounded phase-space neighborhoods, where a compactly supported cutoff can isolate the region of interest. Elliptic and propagation arguments also make assertions that deliberately ignore compact-frequency behavior: a symbol may fail or be irrelevant for small $|\xi|$ but have the decisive property once the covector is sufficiently large. To express that distinction without repeatedly adding cutoffs and radius clauses, the calculus needs a local phrase for properties that hold over a base point only for all sufficiently large fiber variables.
[definition: Symbol At Fiber Infinity]
Let $U\subset \mathbb R^n$ be open, let $x_0\in U$, and let $a(x,\xi;h)$ be a symbol. A statement about $a$ holds at fiber infinity over $x_0$ in $U$ if there exist $R>0$ and an open neighbourhood $U_0\subset U$ of $x_0$ such that the statement holds for all $x\in U_0$, all $\xi\in\mathbb R^n$ with $|\xi|\ge R$, and all $0<h\le h_0$ whenever the statement depends on $h$.
[/definition]
This terminology is a way to separate large-frequency information from compact-frequency information. Later, ellipticity at fiber infinity will mean invertibility of the leading symbol for sufficiently large $|\xi|$ in the relevant region of $x$.
[example: Elliptic Behavior Of The Laplace Symbol At Fiber Infinity]
For $p(x,\xi)=|\xi|^2$, the symbol is independent of $x$ and $h$. Its zeroth derivative is controlled by the order function $\langle\xi\rangle^2$ because
\begin{align*}
|p(x,\xi)|=|\xi|^2\le 1+|\xi|^2=\langle\xi\rangle^2.
\end{align*}
For first fiber derivatives,
\begin{align*}
\partial_{\xi_j}p(x,\xi)=2\xi_j.
\end{align*}
Since $|\xi_j|\le |\xi|\le \langle\xi\rangle\le \langle\xi\rangle^2$, we get
\begin{align*}
|\partial_{\xi_j}p(x,\xi)|\le 2\langle\xi\rangle^2.
\end{align*}
For second fiber derivatives,
\begin{align*}
\partial_{\xi_j}\partial_{\xi_k}p(x,\xi)=2\delta_{jk}.
\end{align*}
Hence
\begin{align*}
|\partial_{\xi_j}\partial_{\xi_k}p(x,\xi)|\le 2\le 2\langle\xi\rangle^2.
\end{align*}
All fiber derivatives of order at least three vanish, and every $x$-derivative of positive order vanishes. Therefore every mixed derivative is bounded by a constant times $\langle\xi\rangle^2$, uniformly in $h$, so $p\in S^2(T^*\mathbb R^n)$.
To check ellipticity at fiber infinity, choose $R=1$. If $|\xi|\ge 1$, then
\begin{align*}
|\xi|^2\ge 1.
\end{align*}
Adding $|\xi|^2$ to both sides gives
\begin{align*}
2|\xi|^2\ge 1+|\xi|^2.
\end{align*}
Dividing by $2$ and using $\langle\xi\rangle^2=1+|\xi|^2$, we obtain
\begin{align*}
p(x,\xi)=|\xi|^2\ge \frac{1}{2}\langle\xi\rangle^2.
\end{align*}
Thus the Laplace symbol has the required lower bound for all sufficiently large $|\xi|$, while no condition is imposed on the bounded-frequency region.
[/example]
The example shows why estimates at fiber infinity deliberately ignore compact-frequency effects. We next need vocabulary for the opposite kind of negligible effect: errors that may exist globally but are smaller than every finite power of $h$ in all symbol seminorms.
[definition: Residual Symbol Family]
Let $m$ be an order function. A family
\begin{align*}
r:T^*\mathbb R^n\times(0,h_0]\longrightarrow \mathbb C
\end{align*}
with $r\in C^\infty(T^*\mathbb R^n\times(0,h_0];\mathbb C)$ is residual relative to $S(m)$ if
\begin{align*}
r\in h^\infty S(m).
\end{align*}
[/definition]
Residual families are not discarded because they vanish pointwise; they are discarded because they are beyond all finite orders in the semiclassical expansion. To use them safely in later algebra, we need the following ideal property showing that multiplication by an ordinary symbol preserves residual size.
[quotetheorem:7293]
[citeproof:7293]
The ideal property relies on one factor being residual and the other factor having symbol seminorm control. If $r=e^{-1/h}\langle\xi\rangle^{m_1}$ and $a_h=e^{1/h}$, then $a_hr=\langle\xi\rangle^{m_1}$ is no longer residual; the failure occurs because $a_h$ is not bounded in any ordinary symbol class uniformly as $h\to0$. Likewise, the theorem does not say that division by a residual family preserves symbolic behavior, since residual functions may vanish or be too small to invert in the symbol topology.
The result is exactly what later error estimates need: once an error is residual, multiplying it by bounded pieces of the calculus cannot bring it back into any finite asymptotic order. The chapter therefore ends with the symbolic hierarchy needed for quantization: ordinary symbols, mildly singular symbols, classical expansions, principal layers, compactly supported localizers, and residual errors. In the next chapter, these functions on $T^*\mathbb R^n$ become operators through semiclassical quantization, and the estimates developed here become mapping and composition estimates.
The symbol classes developed here describe what can be controlled before any operator is formed. The next chapter uses those classes to define semiclassical quantization on Euclidean space, so that symbols on $T^*\mathbb R^n$ become operators acting on the oscillatory families introduced earlier.
# 3. Quantization on Euclidean Space
This chapter turns symbols on phase space into operators acting on functions on Euclidean space. It assumes the semiclassical Fourier transform, oscillatory integrals, Schwartz functions, tempered distributions, and the symbol classes $S^m(T^*\mathbb R^n)$ developed earlier. The guiding question is how a function $a(x,\xi;h)$ on $T^*\mathbb R^n$ should act on an oscillatory function whose frequency is measured at scale $h^{-1}$. Kohn--Nirenberg quantization gives the first answer by placing $x$ before $hD_x$, while Weyl quantization modifies the prescription so that the position and frequency variables are treated symmetrically.
The chapter also explains why quantization is not a single canonical operation at the level of formulas. Different conventions produce operators that agree to leading semiclassical order but differ by lower powers of $h$. This flexibility is essential later, because symbolic calculus, adjoints, real principal symbols, and microlocal cutoffs are often cleaner in different conventions.
## Kohn--Nirenberg Quantization and Operator Kernels
The first problem is to convert the phase-space multiplier $a(x,\xi;h)$ into an operator while retaining the semiclassical Fourier scaling introduced earlier. Since the variable $x$ represents where the output is observed, the Kohn--Nirenberg convention evaluates the symbol at the output point and inserts the input function through its inverse semiclassical Fourier representation.
[definition: Kohn--Nirenberg Quantization]
Let $a \in S^m(T^*\mathbb R^n)$ be a semiclassical symbol. The Kohn--Nirenberg quantization of $a$ is the continuous operator
\begin{align*}
\operatorname{Op}_h(a):\mathcal S(\mathbb R^n)\to \mathcal S'(\mathbb R^n)
\end{align*}
initially defined by
\begin{align*}
\operatorname{Op}_h(a)u(x)
&= \frac{1}{(2\pi h)^n}\int_{\mathbb R^n}\int_{\mathbb R^n} e^{i(x-y)\cdot \xi/h} a(x,\xi;h)u(y)\,dy\,d\xi .
\end{align*}
[/definition]
This formula should be read as an oscillatory integral when $a$ is not integrable in $\xi$. The placement of $a(x,\xi;h)$ at the output point is what distinguishes this convention from the symmetric one introduced later.
[example: Multiplication Symbols]
Let $u\in \mathcal S(\mathbb R^n)$ and set $a(x,\xi;h)=V(x)$. Substituting this symbol into the Kohn--Nirenberg formula gives
\begin{align*}
\operatorname{Op}_h(V)u(x)=\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot \xi/h}V(x)u(y)\,dy\,d\xi.
\end{align*}
Since $V(x)$ is independent of $y$ and $\xi$, it factors out:
\begin{align*}
\operatorname{Op}_h(V)u(x)=V(x)\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot \xi/h}\left(\int_{\mathbb R^n}e^{-iy\cdot \xi/h}u(y)\,dy\right)d\xi.
\end{align*}
The inner parenthesis is the semiclassical Fourier transform of $u$, and semiclassical Fourier inversion gives
\begin{align*}
\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot \xi/h}\left(\int_{\mathbb R^n}e^{-iy\cdot \xi/h}u(y)\,dy\right)d\xi=u(x).
\end{align*}
Therefore
\begin{align*}
\operatorname{Op}_h(V)u(x)=V(x)u(x).
\end{align*}
Thus a symbol depending only on the physical variable produces ordinary multiplication by that function.
[/example]
The preceding example is the anchor case: quantization extends multiplication by $x$-dependent functions and Fourier multipliers by $\xi$-dependent functions into one phase-space operation. To use these operators in estimates, we need a kernel formula that records where the operator is singular and how oscillation controls the off-diagonal region; the next theorem provides that representation.
[quotetheorem:7294]
[citeproof:7294]
The theorem says that semiclassical pseudodifferential operators are microlocally concentrated near the diagonal in physical space, but their singularity along the diagonal remembers the fibre variable $\xi$. The symbol hypothesis is essential: without symbol estimates in $\xi$, repeated integration by parts need not control the off-diagonal integral, and an amplitude such as $e^{|\xi|^2}$ would not define the same kind of oscillatory kernel. The Schwartz input space keeps the formula safely inside the distributional kernel framework; boundedness on $L^2$ or on Sobolev spaces is a separate mapping theorem, not a consequence of this representation alone. This is why phase-space localisation can be studied through kernels while still tracking frequency.
[example: Quantization of Coordinate and Momentum]
Let $u\in\mathcal S(\mathbb R^n)$, and write its semiclassical Fourier transform as
\begin{align*}
\widehat u_h(\xi)=\int_{\mathbb R^n}e^{-iy\cdot \xi/h}u(y)\,dy.
\end{align*}
For the coordinate symbol $a(x,\xi)=x_i$, the Kohn--Nirenberg formula gives
\begin{align*}
\operatorname{Op}_h(x_i)u(x)=x_i\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot \xi/h}\widehat u_h(\xi)\,d\xi.
\end{align*}
By semiclassical Fourier inversion, the integral is $u(x)$, so
\begin{align*}
\operatorname{Op}_h(x_i)u(x)=x_i u(x).
\end{align*}
For the momentum symbol $a(x,\xi)=\xi_j$, the same formula gives
\begin{align*}
\operatorname{Op}_h(\xi_j)u(x)=\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot \xi/h}\xi_j\widehat u_h(\xi)\,d\xi.
\end{align*}
Since $hD_{x_j}=-ih\partial_{x_j}$ and
\begin{align*}
hD_{x_j}e^{ix\cdot \xi/h}=-ih\frac{i\xi_j}{h}e^{ix\cdot \xi/h}=\xi_j e^{ix\cdot \xi/h},
\end{align*}
we may rewrite the preceding integral as
\begin{align*}
\operatorname{Op}_h(\xi_j)u(x)=hD_{x_j}\left(\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot \xi/h}\widehat u_h(\xi)\,d\xi\right).
\end{align*}
Fourier inversion then gives
\begin{align*}
\operatorname{Op}_h(\xi_j)u(x)=hD_{x_j}u(x).
\end{align*}
For the product symbol $a(x,\xi)=x_i\xi_j$, the factor $x_i$ is evaluated at the output point $x$, so it is independent of the integration variables $y$ and $\xi$:
\begin{align*}
\operatorname{Op}_h(x_i\xi_j)u(x)=x_i\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot \xi/h}\xi_j\widehat u_h(\xi)\,d\xi.
\end{align*}
Using the momentum computation just obtained,
\begin{align*}
\operatorname{Op}_h(x_i\xi_j)u(x)=x_i hD_{x_j}u(x).
\end{align*}
Thus Kohn--Nirenberg ordering places the coordinate multiplier before the momentum operator.
[/example]
This example exposes the noncommutativity that quantization must manage. Classically $x_i\xi_j=\xi_jx_i$, while the corresponding operators need not agree because differentiation sees multiplication.
## Weyl Quantization and Symmetric Ordering
The next problem is to choose a convention compatible with adjoints and real-valued symbols. If a classical observable is real, the corresponding quantum operator should be self-adjoint at least at the formal level; this is not built into the Kohn--Nirenberg ordering. Weyl quantization repairs this by evaluating the symbol at the midpoint between input and output points.
[definition: Weyl Quantization]
Let $a \in S^m(T^*\mathbb R^n)$. The Weyl quantization of $a$ is the continuous operator
\begin{align*}
\operatorname{Op}^w_h(a):\mathcal S(\mathbb R^n)\to \mathcal S'(\mathbb R^n)
\end{align*}
initially defined by
\begin{align*}
\operatorname{Op}^w_h(a)u(x)
&=\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot \xi/h}a\left(\frac{x+y}{2},\xi;h\right)u(y)\,dy\,d\xi.
\end{align*}
[/definition]
The midpoint is the algebraic expression of symmetric ordering. It averages the roles of the input and output variables, so the kernel has a built-in symmetry under interchange of $x$ and $y$ combined with complex conjugation of the symbol.
[example: Weyl Symmetrization of x Times Momentum]
Let $u\in\mathcal S(\mathbb R^n)$. Substituting $a(x,\xi)=x_i\xi_j$ into the Weyl formula gives
\begin{align*}
\operatorname{Op}^w_h(x_i\xi_j)u(x)=\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\xi/h}\frac{x_i+y_i}{2}\xi_j u(y)\,dy\,d\xi.
\end{align*}
Splitting the factor $(x_i+y_i)/2$ gives
\begin{align*}
\operatorname{Op}^w_h(x_i\xi_j)u(x)=\frac{x_i}{2}\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot\xi/h}\xi_j\widehat u_h(\xi)\,d\xi+\frac{1}{2}\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot\xi/h}\xi_j\widehat{(x_i u)}_h(\xi)\,d\xi.
\end{align*}
Since $hD_{x_j}=-ih\partial_{x_j}$,
\begin{align*}
hD_{x_j}e^{ix\cdot\xi/h}=-ih\frac{i\xi_j}{h}e^{ix\cdot\xi/h}=\xi_j e^{ix\cdot\xi/h}.
\end{align*}
Applying semiclassical Fourier inversion first to $u$ and then to $x_i u$, the previous display identifies the two integrals as $hD_{x_j}u(x)$ and $hD_{x_j}(x_i u)(x)$. Hence
\begin{align*}
\operatorname{Op}^w_h(x_i\xi_j)u(x)=\frac{1}{2}x_i hD_{x_j}u(x)+\frac{1}{2}hD_{x_j}(x_i u)(x).
\end{align*}
Expanding the derivative in the second term,
\begin{align*}
hD_{x_j}(x_i u)=-ih\partial_{x_j}(x_i u)=-ih(\delta_{ij}u+x_i\partial_{x_j}u).
\end{align*}
Because $x_i hD_{x_j}u=x_i(-ih\partial_{x_j}u)$, this becomes
\begin{align*}
hD_{x_j}(x_i u)=x_i hD_{x_j}u-ih\delta_{ij}u.
\end{align*}
Substituting back,
\begin{align*}
\operatorname{Op}^w_h(x_i\xi_j)u=x_i hD_{x_j}u-\frac{ih}{2}\delta_{ij}u.
\end{align*}
Thus Weyl ordering is the average of placing $x_i$ before and after $hD_{x_j}$, and the order-$h$ correction records the noncommutation of multiplication by $x_i$ with differentiation in $x_j$.
[/example]
The example shows the main pattern of the calculus: different quantizations agree on principal symbols and differ in lower-order terms. For applications to symmetric PDE and real-valued observables, we need a precise adjoint statement rather than only an ordering slogan; the next theorem gives the formal adjoint formula that motivates Weyl quantization.
[quotetheorem:7295]
[citeproof:7295]
The midpoint hypothesis is doing real work here. For example, the Kohn--Nirenberg operator associated to the real symbol $x_i\xi_j$ is $x_i hD_{x_j}$, while its formal adjoint is $hD_{x_j}x_i=x_i hD_{x_j}-ih\delta_{ij}$, so a real Kohn--Nirenberg symbol need not give a formally self-adjoint operator. The Weyl statement is still only formal on $\mathcal S(\mathbb R^n)$: it does not by itself prove essential self-adjointness, choose a closed $L^2$ domain, or give spectral information. For Kohn--Nirenberg quantization, the adjoint is still pseudodifferential, but the symbol is not simply $\overline a$. The asymmetry in $x$ and $y$ forces a correction expansion involving mixed derivatives.
[remark: Kohn--Nirenberg Adjoints]
The formal adjoint of $\operatorname{Op}_h(a)$ has a Kohn--Nirenberg symbol asymptotic to
\begin{align*}
a^*(x,\xi;h)\sim \sum_{\alpha}\frac{h^{|\alpha|}}{\alpha!}D_\xi^\alpha \partial_x^\alpha \overline{a(x,\xi;h)},
\end{align*}
with the convention $D_{\xi_j}=-i\partial_{\xi_j}$. The leading term is $\overline a$, and the remaining terms measure the failure of the quantization to be symmetric.
[/remark]
This asymptotic formula foreshadows the symbolic expansions used in composition. It also explains why the principal symbol is independent of the quantization convention, while subprincipal information depends on a choice.
## General Tau-Quantization and Conversion Between Conventions
The final problem in this chapter is to organise all common orderings into one family. Kohn--Nirenberg and Weyl quantization are two points in a one-parameter family indexed by where the symbol is evaluated between the output point $x$ and input point $y$.
[definition: Tau-Quantization]
Let $\tau\in\mathbb R$ and let $a\in S^m(T^*\mathbb R^n)$. The $\tau$-quantization of $a$ is the continuous operator
\begin{align*}
\operatorname{Op}_{h,\tau}(a):\mathcal S(\mathbb R^n)\to\mathcal S'(\mathbb R^n)
\end{align*}
defined by
\begin{align*}
\operatorname{Op}_{h,\tau}(a)u(x)
=\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\xi/h}a((1-\tau)x+\tau y,\xi;h)u(y)\,dy\,d\xi.
\end{align*}
[/definition]
The cases $\tau=0$ and $\tau=1/2$ recover Kohn--Nirenberg and Weyl quantization respectively. The parameter $\tau$ is not a new analytic structure; it is a bookkeeping device that records an ordering convention.
[example: Three Orderings of x Times Momentum]
Let $u\in\mathcal S(\mathbb R^n)$ and set
\begin{align*}
\widehat u_h(\xi)=\int_{\mathbb R^n}e^{-iy\cdot \xi/h}u(y)\,dy.
\end{align*}
Substituting $a(x,\xi)=x_i\xi_j$ into the $\tau$-quantization formula gives
\begin{align*}
\operatorname{Op}_{h,\tau}(x_i\xi_j)u(x)=\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\xi/h}\bigl((1-\tau)x_i+\tau y_i\bigr)\xi_j u(y)\,dy\,d\xi.
\end{align*}
Splitting the two terms in the amplitude,
\begin{align*}
\operatorname{Op}_{h,\tau}(x_i\xi_j)u(x)=(1-\tau)x_i\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot\xi/h}\xi_j\widehat u_h(\xi)\,d\xi+\tau\frac{1}{(2\pi h)^n}\int_{\mathbb R^n}e^{ix\cdot\xi/h}\xi_j\widehat{(x_i u)}_h(\xi)\,d\xi.
\end{align*}
Since $hD_{x_j}=-ih\partial_{x_j}$,
\begin{align*}
hD_{x_j}e^{ix\cdot\xi/h}=-ih\frac{i\xi_j}{h}e^{ix\cdot\xi/h}=\xi_j e^{ix\cdot\xi/h}.
\end{align*}
Semiclassical Fourier inversion therefore identifies the first integral as $hD_{x_j}u(x)$ and the second as $hD_{x_j}(x_i u)(x)$. Hence
\begin{align*}
\operatorname{Op}_{h,\tau}(x_i\xi_j)u(x)=(1-\tau)x_i hD_{x_j}u(x)+\tau hD_{x_j}(x_i u)(x).
\end{align*}
Now expand the product derivative:
\begin{align*}
hD_{x_j}(x_i u)=-ih\partial_{x_j}(x_i u)=-ih(\delta_{ij}u+x_i\partial_{x_j}u).
\end{align*}
Since $x_i hD_{x_j}u=-ihx_i\partial_{x_j}u$, this is
\begin{align*}
hD_{x_j}(x_i u)=x_i hD_{x_j}u-ih\delta_{ij}u.
\end{align*}
Substituting this into the previous expression gives
\begin{align*}
\operatorname{Op}_{h,\tau}(x_i\xi_j)u(x)=(1-\tau)x_i hD_{x_j}u(x)+\tau x_i hD_{x_j}u(x)-i\tau h\delta_{ij}u(x).
\end{align*}
Combining the two $x_i hD_{x_j}u$ terms,
\begin{align*}
\operatorname{Op}_{h,\tau}(x_i\xi_j)u=x_i hD_{x_j}u-i\tau h\delta_{ij}u.
\end{align*}
Thus $\tau=0$ gives the Kohn--Nirenberg ordering $x_i hD_{x_j}$, $\tau=1/2$ gives the Weyl operator $x_i hD_{x_j}-(ih/2)\delta_{ij}$, and $\tau=1$ gives $hD_{x_j}(x_i u)$, meaning that the momentum operator acts after multiplication by $x_i$.
[/example]
Since the same operator can be represented by different symbols under different choices of $\tau$, we need an explicit conversion rule. The rule is a Taylor expansion in the base variable, with powers of $h$ supplied by integration by parts in the dual variable.
[quotetheorem:7296]
[citeproof:7296]
The same argument works between any two $\tau$-quantizations. If $a_\tau$ and $a_\sigma$ represent the same operator in the $\tau$ and $\sigma$ conventions, then their relationship is governed by an exponential of $(\sigma-\tau)ih\sum_j\partial_{x_j}\partial_{\xi_j}$ when converting from the $\tau$-symbol to the $\sigma$-symbol. The symbol-class hypothesis is needed for the Taylor remainder and Borel summation steps; without it, the formal series need not define a controlled lower-order correction. The equality is also convention-dependent rather than a unique pointwise identity of symbols: the example $x_i\xi_j$ already shows that Weyl and Kohn--Nirenberg symbols differ by $-(ih/2)\delta_{ij}$ while defining the same operator. Thus the theorem identifies operators modulo the chosen symbolic calculus, not an intrinsic symbol representative independent of quantization.
[remark: Principal Symbols Are Convention Independent]
Quantization choices affect terms below the leading semiclassical order. Hence the principal symbol of a semiclassical pseudodifferential operator is well-defined modulo lower-order symbols once a symbol class filtration has been fixed. Subprincipal symbols require extra convention data, and Weyl quantization is often chosen because it makes the formal adjoint and real-symbol conditions cleaner.
[/remark]
This completes the passage from symbols to operators on $\mathbb R^n$. The next stage is to compose these operators and compute how the product of two quantized symbols reflects the symplectic geometry of $T^*\mathbb R^n$.
Once symbols have been quantized, the natural question is how these operators interact. The next chapter answers that by developing the semiclassical pseudodifferential calculus, where composition, adjoints, and asymptotic expansions reflect the geometry of phase space.
# 4. The Semiclassical Pseudodifferential Calculus
This chapter turns symbol estimates into an algebra of operators. In the previous chapters, quantization assigned an operator to a symbol and basic mapping estimates explained why symbol order controls analytic strength. The next question is structural: if two semiclassical pseudodifferential operators are composed, commuted, or adjointed, what happens to their symbols, and how accurately does the answer determine the operator?
The answer is the semiclassical pseudodifferential calculus. Products of operators become asymptotic products of symbols, commutators become Poisson brackets to first order in $h$, adjoints are governed by conjugation plus lower-order corrections, and operators with rapidly decaying symbols form a residual class. These rules are the bookkeeping system behind microlocal elliptic parametrices, propagation estimates, and later spectral constructions.
## Composition and the Symbolic Product
The first problem is whether quantization respects multiplication. If $a(x,\theta)$ and $b(y,\theta)$ are amplitudes in an oscillatory integral, the product $\theta$-integration and the $y$-integration should localise near $y=x$ and transfer derivatives between $a$ and $b$. The calculus records this localisation as an asymptotic expansion in powers of $h$.
[definition: Semiclassical Symbolic Product]
For symbol orders $m,m' \in \mathbb R$, the semiclassical symbolic product is a bilinear operation, defined modulo residual symbol equivalence,
\begin{align*}
\# : S^m(T^*\mathbb R^n) \times S^{m'}(T^*\mathbb R^n) \to S^{m+m'}(T^*\mathbb R^n),
\end{align*}
whose value on $(a,b)$ is denoted
\begin{align*}
a\# b \in S^{m+m'}(T^*\mathbb R^n)/S^{-\infty}_{\mathrm{res}}(T^*\mathbb R^n).
\end{align*}
[/definition]
This definition only names the formal product class; the fact that it represents operator composition is a theorem of the calculus. For the standard left quantization on $\mathbb R^n$, the next task is to prove the composition law and compute the coefficients of $a\# b$. The expansion has a direct form because derivatives in $\xi$ fall on the first symbol and derivatives in $x$ fall on the second.
[quotetheorem:7297]
[citeproof:7297]
The leading term is ordinary multiplication. The first correction measures the failure of quantization to be an algebra homomorphism, and the $h$-gain is the analytic reason that lower commutator terms are weaker in semiclassical estimates. The symbol hypotheses are essential: if $a$ or $b$ has uncontrolled growth or oscillates on the scale $h^{-1}$ without belonging to a semiclassical symbol class, the integrations by parts used in the proof no longer give uniform seminorm bounds for the remainder. Proper support, or an equivalent cutoff convention, is also part of the background because an arbitrary oscillatory kernel can fail to define a continuous operator on compactly supported test functions after composition. The theorem does not say that $a\# b$ is an exact finite formula; it gives an asymptotic expansion, and changing the quantization changes lower-order coefficients while preserving the principal product. A concrete two-term expansion is the fastest way to see which derivatives matter in computations.
[example: First Terms of a Left-Quantized Product]
Let $a,b \in S^0(T^*\mathbb R^n)$ and use left quantization. In the expansion for $a\# b$, the multi-index $\alpha=0$ contributes
\begin{align*}
\frac{h^0}{i^0 0!}(\partial_\xi^0 a)(\partial_x^0 b)=ab.
\end{align*}
For $|\alpha|=1$, the only possibilities are $\alpha=e_j$ for $1\le j\le n$, so $\alpha!=1$ and
\begin{align*}
\sum_{|\alpha|=1}\frac{h^{|\alpha|}}{i^{|\alpha|}\alpha!}(\partial_\xi^\alpha a)(\partial_x^\alpha b)=\frac{h}{i}\sum_{j=1}^n(\partial_{\xi_j}a)(\partial_{x_j}b).
\end{align*}
Stopping the asymptotic expansion after these terms gives
\begin{align*}
a\# b=ab+\frac{h}{i}\sum_{j=1}^n(\partial_{\xi_j}a)(\partial_{x_j}b)+h^2r_2,
\end{align*}
where $r_2\in S^{-2}$ under the standard convention that each $\xi$-derivative lowers symbol order by one.
Thus $\operatorname{Op}_h(a)\operatorname{Op}_h(b)$ has principal symbol $ab$, and the first correction differentiates the first symbol only in momentum and the second symbol only in position. If $a=a(x)$ is independent of $\xi$, then for every nonzero multi-index $\alpha$ one has $\partial_\xi^\alpha a=0$, so every positive-order correction term vanishes and $a\# b=ab$ for left quantization.
[/example]
The preceding theorem is local in coordinates. On a manifold the same formula is read in charts, with cutoffs inserted to localise kernels and with lower-order changes appearing under coordinate transformations.
[remark: Coordinate Dependence of the Full Product]
The principal product $ab$ is invariantly defined on $T^*M$. The full expansion depends on the chosen quantization and on local coordinates, but different choices give operators whose symbols are related by an asymptotic change of variables. For most microlocal arguments, the invariant data are the principal symbol and the first antisymmetric correction, which is governed by the canonical symplectic structure.
[/remark]
## Weyl Quantization and the Moyal Product
The left calculus is efficient for composition estimates, but it treats $x$ and $\xi$ asymmetrically. Weyl quantization is designed to restore symmetry by evaluating the symbol at the midpoint of the kernel. The problem is then to identify the corresponding product and to see why the Poisson bracket appears with its natural coefficient.
[definition: Weyl Quantization]
For $a \in S^m(T^*\mathbb R^n)$, the semiclassical Weyl quantization is the operator
\begin{align*}
\operatorname{Op}_h^w(a):\mathcal S(\mathbb R^n)\to \mathcal S'(\mathbb R^n)
\end{align*}
defined by
\begin{align*}
\operatorname{Op}_h^w(a)u(x) = \frac{1}{(2\pi h)^n}\int_{\mathbb R^n}\int_{\mathbb R^n} e^{i(x-y)\cdot \xi/h}\, a\left(\frac{x+y}{2},\xi\right)u(y)\,dy\,d\xi,
\end{align*}
for $u \in \mathcal S(\mathbb R^n)$, interpreted as an oscillatory integral.
[/definition]
The midpoint convention makes adjoints cleaner and places commutators in geometric form. To state the first correction invariantly, we need the bracket associated with the canonical symplectic structure on $T^*\mathbb R^n$. This bracket measures how the Hamiltonian vector field of one symbol differentiates the other.
[definition: Poisson Bracket]
The Poisson bracket is the bilinear map
\begin{align*}
\{\cdot,\cdot\}:C^\infty(T^*\mathbb R^n)\times C^\infty(T^*\mathbb R^n)\to C^\infty(T^*\mathbb R^n)
\end{align*}
defined, for $a,b \in C^\infty(T^*\mathbb R^n)$, by
\begin{align*}
\{a,b\} = \sum_{j=1}^n \left(\partial_{\xi_j}a\,\partial_{x_j}b - \partial_{x_j}a\,\partial_{\xi_j}b\right).
\end{align*}
[/definition]
This sign convention is the one compatible with the commutator formula in this course: $[\operatorname{Op}_h^w(a),\operatorname{Op}_h^w(b)]$ has principal term $(h/i)\operatorname{Op}_h^w(\{a,b\})$. The bracket describes only the first antisymmetric correction, while composing two Weyl operators requires all symmetric and antisymmetric corrections together. To track the full Weyl product before passing to commutators, we introduce a dedicated product on symbols.
[definition: Moyal Product]
For symbol orders $m,m' \in \mathbb R$, the Moyal product is a bilinear operation, defined modulo residual symbol equivalence,
\begin{align*}
\#_w:S^m(T^*\mathbb R^n)\times S^{m'}(T^*\mathbb R^n)\to S^{m+m'}(T^*\mathbb R^n).
\end{align*}
For $a \in S^m(T^*\mathbb R^n)$ and $b \in S^{m'}(T^*\mathbb R^n)$, it is characterized by
\begin{align*}
\operatorname{Op}_h^w(a)\operatorname{Op}_h^w(b)=\operatorname{Op}_h^w(a\#_w b)
\end{align*}
modulo residual operators.
[/definition]
The Moyal product packages all correction terms through the bidifferential operator associated with the symplectic form. After defining it abstractly, the main computational need is to isolate a consequence that survives invariantly in dynamics: the antisymmetric first correction. In a Weyl commutator the symmetric part cancels, leaving the Poisson bracket as the leading term.
[quotetheorem:7299]
[citeproof:7299]
This theorem explains why Weyl quantization is preferred when self-adjointness and Hamiltonian dynamics matter. Symmetric ordering cancels the terms that would otherwise obscure the geometric bracket, while the first antisymmetric term remains visible at order $h$. The symbol assumptions are needed because the calculation is controlled by uniform symbol seminorms. A concrete failure is the family $a_h(x,\xi)=e^{ix_1/h}\chi(x,\xi)$ with $\chi\in C_c^\infty(T^*\mathbb R^n)$ nonzero. Although $a_h$ is smooth and compactly supported for each fixed $h$, its derivatives satisfy $\partial_{x_1}^k a_h=h^{-k}e^{ix_1/h}\chi+\cdots$, so the usual $S^0$ seminorms are not uniformly bounded. The theorem also does not make the Weyl product commutative: it identifies the controlled leading failure of commutativity. Quadratic symbols give a compact illustration because the symbolic expansions stop after finitely many derivatives.
[example: Harmonic Oscillator and Weyl Ordering]
Consider $a(x,\xi)=|\xi|^2+|x|^2$ on $T^*\mathbb R^n$. Since $|\xi|^2=\sum_{j=1}^n \xi_j^2$ is independent of $x$, its Weyl quantization agrees with the semiclassical Fourier multiplier $\sum_{j=1}^n(hD_{x_j})^2$. Since $|x|^2=\sum_{j=1}^n x_j^2$ is independent of $\xi$, its Weyl quantization is multiplication by $\sum_{j=1}^n x_j^2$. By linearity of $\operatorname{Op}_h^w$,
\begin{align*}
\operatorname{Op}_h^w(|\xi|^2+|x|^2)=\sum_{j=1}^n\left((hD_{x_j})^2+x_j^2\right).
\end{align*}
For the mixed quadratic symbol $x_j\xi_j$, the midpoint in the Weyl kernel symmetrizes multiplication by $x_j$ and the differential operator $hD_{x_j}$, so
\begin{align*}
\operatorname{Op}_h^w(x_j\xi_j)=\frac{1}{2}\left(x_jhD_{x_j}+hD_{x_j}x_j\right).
\end{align*}
Applying this operator to $u\in\mathcal S(\mathbb R^n)$ and using $D_{x_j}=\frac{1}{i}\partial_{x_j}$ gives
\begin{align*}
\frac{1}{2}\left(x_jhD_{x_j}+hD_{x_j}x_j\right)u=\frac{1}{2}\left(x_j\frac{h}{i}\partial_{x_j}u+\frac{h}{i}\partial_{x_j}(x_ju)\right).
\end{align*}
The product rule gives $\partial_{x_j}(x_ju)=u+x_j\partial_{x_j}u$, hence
\begin{align*}
\frac{1}{2}\left(x_jhD_{x_j}+hD_{x_j}x_j\right)u=x_j\frac{h}{i}\partial_{x_j}u+\frac{h}{2i}u.
\end{align*}
Therefore
\begin{align*}
\operatorname{Op}_h^w(x_j\xi_j)=x_jhD_{x_j}+\frac{h}{2i}.
\end{align*}
The lower-order constant is exactly the cost of moving $hD_{x_j}$ past multiplication by $x_j$, so quadratic Hamiltonians can differ by order-$h$ constants under different ordering conventions.
[/example]
The example shows that ordering choices can create lower-order terms even when the classical symbol is simple. For estimates and dynamics, however, the antisymmetric part of the product is the more stable object. The Weyl commutator formula above depends on Weyl symmetry in a real way. Without the midpoint quantization, the principal Poisson bracket term is still present, but the next remainder generally appears already at order $h^2$ rather than $h^3$. The theorem also assumes that both operators are in the symbolic calculus; for arbitrary unbounded operators a commutator may fail to preserve the natural domain or may not be represented by any symbol. This limitation is exactly why the next section packages the operators into classes $\Psi_h^m$ with controlled composition, adjoints, and residual errors.
## Commutators, Adjoints, and the Algebra $\Psi_h^m$
The calculus becomes useful when it is stable under the operations that arise in estimates: composition, commutation, taking adjoints, and discarding smoothing errors. The guiding question is which classes of operators are closed under these operations and which part of the symbol survives as invariant leading data.
[definition: Semiclassical Pseudodifferential Class]
Let $M$ be a smooth manifold. The class $\Psi_h^m(M)$ consists of properly supported continuous operators
\begin{align*}
A:C_c^\infty(M)\to C^\infty(M)
\end{align*}
that, in every coordinate chart and after insertion of compactly supported cutoffs, can be written as semiclassical quantizations of symbols in $S^m(T^*M)$ modulo residual operators.
[/definition]
This definition suppresses the partition-of-unity construction, but the point is structural: the operator class is local, while principal symbols glue globally on the cotangent bundle. Once the class is defined, the first closure property to verify is composition, since parametrices and energy estimates both require products of pseudodifferential operators to remain in the calculus. The Euclidean composition theorem gives the local statement, and the theorem below records the global version.
[quotetheorem:7300]
[proofunderconstruction:7300]
The algebra property says that products stay inside the calculus and that their leading symbols multiply. Proper support and pseudodifferential membership are not cosmetic assumptions. On $\mathbb R^n$, the pullback operator $Tu(x)=u(2x)$ is continuous from $C_c^\infty(\mathbb R^n)$ to $C^\infty(\mathbb R^n)$, but its kernel is supported on $y=2x$, not on the diagonal relation used by pseudodifferential kernels. It is a Fourier integral operator associated with a nontrivial canonical graph rather than an element of $\Psi_h^0$, so composing it with a pseudodifferential operator generally transports singularities by dilation instead of preserving the same cotangent base point. This model failure explains why arbitrary continuous operators cannot be admitted into the algebra. The theorem also does not identify all lower-order terms invariantly; those depend on quantization choices and coordinate constructions, while the principal symbol is the stable global object. The next operation is the commutator, where the leading product cancels and a subtler invariant appears. This is the point where the symplectic geometry of $T^*M$ enters the global calculus.
[quotetheorem:7301]
[citeproof:7301]
The formula is abstract, but its normalization is fixed by the basic commutator of a semiclassical derivative with multiplication. The assumption that $A$ and $B$ have principal symbols in the calculus is essential because the cancellation of $ab$ against $ba$ happens at the symbolic level; an arbitrary pair of operators may have a commutator with no controlled order drop. The statement also records only the principal symbol of the commutator as an element of $h\Psi_h^{m+m'-1}(M)$, not a complete invariant formula for every lower coefficient. This computation is also the model for how differential operators sit inside the pseudodifferential calculus.
[example: Computing the Basic Commutator]
Let $D_{x_i}=\frac{1}{i}\partial_{x_i}$, and let $f\in C^\infty(\mathbb R^n)$ act by multiplication. For $u\in\mathcal S(\mathbb R^n)$, the commutator acts by
\begin{align*}
[hD_{x_i},f]u=hD_{x_i}(fu)-f\,hD_{x_i}u.
\end{align*}
Since $D_{x_i}=\frac{1}{i}\partial_{x_i}$, this is
\begin{align*}
[hD_{x_i},f]u=\frac{h}{i}\partial_{x_i}(fu)-f\frac{h}{i}\partial_{x_i}u.
\end{align*}
The product rule gives $\partial_{x_i}(fu)=(\partial_{x_i}f)u+f\partial_{x_i}u$, so
\begin{align*}
[hD_{x_i},f]u=\frac{h}{i}\left((\partial_{x_i}f)u+f\partial_{x_i}u\right)-\frac{h}{i}f\partial_{x_i}u.
\end{align*}
Cancelling the two copies of $\frac{h}{i}f\partial_{x_i}u$ gives
\begin{align*}
[hD_{x_i},f]u=\frac{h}{i}(\partial_{x_i}f)u.
\end{align*}
The symbol of $hD_{x_i}$ is $\xi_i$, while the symbol of multiplication by $f$ is $f(x)$. With the chapter's Poisson bracket convention,
\begin{align*}
\{\xi_i,f\}=\sum_{j=1}^n\left(\partial_{\xi_j}\xi_i\,\partial_{x_j}f-\partial_{x_j}\xi_i\,\partial_{\xi_j}f\right).
\end{align*}
Here $\partial_{\xi_j}\xi_i=\delta_{ij}$, $\partial_{x_j}\xi_i=0$, and $\partial_{\xi_j}f=0$, hence
\begin{align*}
\{\xi_i,f\}=\sum_{j=1}^n\delta_{ij}\partial_{x_j}f=\partial_{x_i}f.
\end{align*}
Thus the operator identity is exactly the symbolic prediction
\begin{align*}
[hD_{x_i},f]=\frac{h}{i}\,\partial_{x_i}f,
\end{align*}
and no lower-order correction remains in this affine-in-$\xi$ and multiplication-in-$x$ case.
[/example]
Adjoints are the next operation needed for energy estimates. The formal adjoint theorem for Weyl quantization proved earlier gives the cleanest model: the principal symbol behaves as expected under complex conjugation, while the lower corrections depend on the chosen quantization convention. In the global calculus the same kernel operation must be combined with proper support and coordinate localization. If the kernel is not properly supported, taking the adjoint can move mass into regions where the local coordinate representation no longer gives uniform estimates on compact sets. If the amplitude has uncontrolled derivatives, conjugating the kernel still defines a formal expression, but the Taylor remainder may fail to be a lower-order symbol. Symbolic adjoint formulas also do not settle self-adjointness as an operator-theoretic domain question; essential self-adjointness and boundary-domain issues require separate analysis.
The term residual means more than lower order. Residual operators are negligible to every algebraic order in $h$ and have smooth kernels with strong bounds, so they may be dropped in symbolic computations until a final estimate requires them. This residual bookkeeping is the mechanism that lets all preceding formulas be used without tracking every smoothing kernel separately.
[definition: Residual Operator]
A family $R_h:C_c^\infty(M)\to C^\infty(M)$ is residual if its Schwartz kernel is smooth and, in every pair of coordinate charts and for all multi-indices $\alpha,\beta$ and all $N\in\mathbb N$, there is a constant $C_{\alpha\beta N}>0$ such that
\begin{align*}
|\partial_x^\alpha\partial_y^\beta K_{R_h}(x,y)|\le C_{\alpha\beta N}h^N
\end{align*}
for $0<h\le 1$ on compact coordinate sets.
[/definition]
The definition is stated in kernel language, but in calculations residual terms usually arise from symbols. To use residual errors safely, we need a criterion that converts rapid symbolic decay into rapid kernel decay and conversely identifies when a smoothing symbol is negligible to every algebraic order in $h$.
[quotetheorem:7298]
[proofunderconstruction:7298]
Residual operators are the smoothing errors of the semiclassical calculus. The next criterion is often the practical way to recognise them from their symbols, especially after an asymptotic construction has produced a symbol whose seminorms decay faster than every power of $h$.
[quotetheorem:7302]
[citeproof:7302]
The criterion distinguishes two notions that are often conflated: finite smoothing and residual smoothing. Both rapid decay in $\xi$ and rapid decay in $h$ are needed: a fixed smoothing symbol such as $\langle\xi\rangle^{-K}\chi(x)$ may improve regularity, but it is not residual because its kernel does not gain arbitrary powers of $h$. Conversely, a factor $h^N$ for one fixed $N$ gives only finite semiclassical smallness, not negligibility to every algebraic order. The converse also depends on proper support; without it, a smooth rapidly decaying local kernel can fail to patch into a uniformly controlled global symbol. The next example separates these behaviours by varying both decay in $\xi$ and decay in $h$.
[example: A Rapidly Decaying Smoothing Symbol]
Let $a_h(x,\xi)=h^N\langle \xi\rangle^{-K}\chi(x)$, where $N,K$ are fixed and $\chi\in C_c^\infty(\mathbb R^n)$. For multi-indices $\alpha,\beta$,
\begin{align*}
\partial_x^\alpha\partial_\xi^\beta a_h(x,\xi)=h^N(\partial_x^\alpha\chi)(x)\,\partial_\xi^\beta\langle \xi\rangle^{-K}.
\end{align*}
Since $\chi$ is smooth with compact support and $\partial_\xi^\beta\langle\xi\rangle^{-K}$ is bounded by a constant times $\langle\xi\rangle^{-K-|\beta|}$, there is a constant $C_{\alpha\beta}$ such that
\begin{align*}
|\partial_x^\alpha\partial_\xi^\beta a_h(x,\xi)|\le C_{\alpha\beta}h^N\langle\xi\rangle^{-K-|\beta|}.
\end{align*}
Thus the symbol seminorms have an $h^N$ gain, but only with the fixed $\xi$-decay supplied by $\langle\xi\rangle^{-K}$.
This is not residual in general. For example,
\begin{align*}
a_h(x,0)=h^N\chi(x).
\end{align*}
If $\chi(x_0)\ne 0$, then $|a_h(x_0,0)|=|\chi(x_0)|h^N$, which is not $O(h^M)$ for every $M$ as $h\to 0$. Also, if $L>K$, then
\begin{align*}
\langle\xi\rangle^L |a_h(x,\xi)|=h^N|\chi(x)|\langle\xi\rangle^{L-K},
\end{align*}
which is unbounded in $\xi$ wherever $\chi(x)\ne 0$. So the example gives finite smoothing and finite semiclassical smallness, not residual smoothing.
By contrast, let $b_h(x,\xi)=q(h)\rho(\xi)\chi(x)$, where $\rho\in S^{-\infty}(\mathbb R^n_\xi)$ and $|q(h)|\le C_Mh^M$ for every $M$. Then for all $\alpha,\beta,L,M$,
\begin{align*}
\sup_{x,\xi}\langle\xi\rangle^L|\partial_x^\alpha\partial_\xi^\beta b_h(x,\xi)|\le C_{\alpha\beta LM}h^M.
\end{align*}
These are exactly rapidly decaying $S^{-\infty}$ seminorms, so the corresponding quantized operator is residual. The distinction is that a large fixed power of $h$ and a large fixed power of $\langle\xi\rangle^{-1}$ do not replace decay to every order in both $h$ and $\xi$.
[/example]
The chapter's calculus can now be summarised as an algebraic dictionary. Products correspond to symbolic products, commutators correspond first to Poisson brackets, adjoints correspond first to conjugation, and residual operators are invisible at the level of finite-order asymptotic expansions. This dictionary is the technical foundation for microlocal ellipticity: the next stage uses it to construct parametrices and to identify where a distribution is singular in phase space.
The calculus now tells us how operators compose, but not yet how large they are on the spaces where solutions live. The next chapter converts the symbolic rules into mapping properties on semiclassical Sobolev spaces, providing the analytic estimates needed for elliptic arguments later on.
# 5. Mapping Properties and Semiclassical Sobolev Spaces
This chapter turns the symbolic calculus into estimates on function spaces. The preceding chapters explained how semiclassical quantization turns a phase-space function $a(x,\xi;h)$ into an operator $\operatorname{Op}_h(a)$, and how composition and adjoints are controlled by symbol seminorms. We now ask how those operators act on semiclassical Sobolev spaces, how much regularity an order-$m$ operator loses, and why singularities can only be moved in phase space by the part of the operator whose kernel is not smooth.
## Semiclassical Sobolev Spaces
The basic difficulty is that the family $e^{ix\cdot \xi_0/h}$ has derivatives of size $h^{-1}$ while representing a bounded semiclassical frequency. Standard Sobolev norms see this as increasingly rough as $h\to 0$, so we need a scale of spaces in which $hD_x$ rather than $D_x$ is the unit derivative.
[definition: Semiclassical Sobolev Space]
For $s\in\mathbb R$ and $0<h\le h_0$, the semiclassical Sobolev space $H^s_h(\mathbb R^n)$ is the vector space $H^s(\mathbb R^n)$ equipped with the norm
\begin{align*}
\|u\|_{H^s_h} = \|\langle hD\rangle^s u\|_{L^2}, \qquad \langle hD\rangle^s = \operatorname{Op}_h(\langle \xi\rangle^s),
\end{align*}
where $\langle \xi\rangle=(1+|\xi|^2)^{1/2}$.
[/definition]
For each fixed $h>0$, this is the same vector space as the usual Sobolev space $H^s(\mathbb R^n)$, but the norm changes with $h$. The point of the definition is uniformity: estimates should remain bounded as $h\to 0$ when the input has semiclassical, rather than classical, regularity. Before asking for uniform estimates, one first checks that no new vector space has been introduced at a fixed parameter and that the comparison constants are allowed to remember the chosen value of $h$.
[quotetheorem:7303]
[citeproof:7303]
This theorem is not meant to give a uniform comparison as $h\to 0$. The constants must depend on $h$: for instance, if $u_h(x)=e^{ix\cdot\xi_0/h}\chi(x)$ with $\xi_0\ne 0$, then $\|u_h\|_{H^s}$ grows like a power of $h^{-1}$ for $s>0$, while $\|u_h\|_{H^s_h}$ remains bounded at the semiclassical frequency scale. Thus the fixed-$h$ hypothesis is necessary, and the theorem should be read as a topological comparison rather than a uniform estimate. Its forward use is to justify working on the familiar vector space $H^s(\mathbb R^n)$ while measuring size with the $h$-dependent norm that matches the calculus.
[example: Plane Wave With Fixed Semiclassical Frequency]
Let $\chi\in C_c^\infty(\mathbb R^n)$ and set $u_h(x)=e^{ix\cdot \xi_0/h}\chi(x)$, with $\xi_0$ fixed. If $\xi_{0,j}\ne 0$, then
\begin{align*}
\partial_{x_j}u_h(x)=e^{ix\cdot \xi_0/h}\left(\frac{i\xi_{0,j}}{h}\chi(x)+\partial_{x_j}\chi(x)\right).
\end{align*}
Equivalently,
\begin{align*}
h\,\partial_{x_j}u_h(x)=e^{ix\cdot \xi_0/h}\left(i\xi_{0,j}\chi(x)+h\,\partial_{x_j}\chi(x)\right).
\end{align*}
Since multiplication by $e^{ix\cdot \xi_0/h}$ preserves $L^2$ norms,
\begin{align*}
h\|\partial_{x_j}u_h\|_{L^2}\ge |\xi_{0,j}|\|\chi\|_{L^2}-h\|\partial_{x_j}\chi\|_{L^2}.
\end{align*}
Thus, for all sufficiently small $h$, $\|\partial_{x_j}u_h\|_{L^2}\ge c h^{-1}$ for some $c>0$ whenever $\xi_{0,j}\ne 0$, so the usual Sobolev norms detect the oscillation as high classical frequency.
The semiclassical derivatives see the same family differently. With $D_{x_j}=-i\partial_{x_j}$,
\begin{align*}
hD_{x_j}u_h=e^{ix\cdot \xi_0/h}\left(\xi_{0,j}\chi+hD_{x_j}\chi\right).
\end{align*}
For a multi-index $\alpha$, repeated use of the product rule gives
\begin{align*}
(hD_x)^\alpha u_h=e^{ix\cdot \xi_0/h}\sum_{\beta\le \alpha}\binom{\alpha}{\beta}\xi_0^{\alpha-\beta}h^{|\beta|}D_x^\beta\chi.
\end{align*}
Hence, for each integer $k\ge 0$,
\begin{align*}
\sum_{|\alpha|\le k}\|(hD_x)^\alpha u_h\|_{L^2}\le C_{k,\xi_0,h_0}\sum_{|\beta|\le k}\|D_x^\beta\chi\|_{L^2}.
\end{align*}
Equivalently, using the Fourier transform,
\begin{align*}
\widehat{u_h}(\zeta)=\widehat{\chi}\left(\zeta-\frac{\xi_0}{h}\right).
\end{align*}
Therefore
\begin{align*}
\|u_h\|_{H^s_h}^2=\int_{\mathbb R^n}\langle h\zeta\rangle^{2s}\left|\widehat{\chi}\left(\zeta-\frac{\xi_0}{h}\right)\right|^2\,d\zeta.
\end{align*}
After the change of variables $\eta=\zeta-\xi_0/h$, this becomes
\begin{align*}
\|u_h\|_{H^s_h}^2=\int_{\mathbb R^n}\langle \xi_0+h\eta\rangle^{2s}|\widehat{\chi}(\eta)|^2\,d\eta.
\end{align*}
Since $\widehat{\chi}$ is rapidly decaying and $0<h\le h_0$, this integral is bounded by a constant depending only on finitely many fixed Schwartz seminorms of $\chi$, $s$, $\xi_0$, and $h_0$. The oscillation is large for the ordinary derivative $D_x$, but it is a bounded frequency for the semiclassical derivative $hD_x$.
[/example]
The example shows that the correct derivative count is carried by the ordinary Fourier multiplier $\langle h\zeta\rangle^s$, where $\zeta$ denotes the usual Fourier variable dual to $x$. In semiclassical notation this same operator is written $\operatorname{Op}_h(\langle \xi\rangle^s)$, because the quantization convention inserts the phase $e^{i(x-y)\cdot \xi/h}$ and therefore identifies the symbol variable $\xi$ with the scaled frequency $h\zeta$. To move regularity from the input side of an estimate to the output side, we package this multiplier as an operator that raises or lowers semiclassical Sobolev order.
[definition: Semiclassical Order Reduction Operator]
For $s\in\mathbb R$, the order reduction operator of order $s$ is the semiclassical pseudodifferential operator
\begin{align*}
\Lambda_h^s = \langle hD\rangle^s=\operatorname{Op}_h(\langle \xi\rangle^s):\mathcal S(\mathbb R^n)\to\mathcal S'(\mathbb R^n).
\end{align*}
[/definition]
The definition names the multiplier, but naming it is not enough for estimates: we need to know exactly how applying $\Lambda_h^s$ changes the Sobolev index. The preceding definition of $H^r_h$ measures $r$ semiclassical derivatives by applying $\langle hD\rangle^r$; composing with $\Lambda_h^s$ should therefore consume $s$ of those derivatives and land in order $r-s$. The next theorem makes that bookkeeping precise and supplies the order-shifting device used later to reduce general Sobolev mapping estimates to $L^2$ boundedness.
[quotetheorem:7304]
[citeproof:7304]
This device is used repeatedly below: prove an $L^2$ estimate after conjugating by $\Lambda_h^s$ and $\Lambda_h^{m-s}$. The target order cannot be improved in general. For a wave packet $u_h(x)=e^{ix\cdot \xi_0/h}\chi(x)$ with $|\xi_0|$ large but fixed and $\chi\in C_c^\infty(\mathbb R^n)$, applying $\Lambda_h^s$ multiplies the leading oscillatory component by $\langle \xi_0\rangle^s$; it changes the Sobolev count by exactly $s$ rather than producing any smoothing beyond that. Thus $\Lambda_h^s$ is an isomorphism from $H^r_h$ onto $H^{r-s}_h$, not a map into a better target space such as $H^{r-s+\varepsilon}_h$ uniformly over high semiclassical frequencies. It also connects this chapter to the familiar elliptic-regularity principle from PDE: a positive elliptic multiplier measures regularity, while its inverse trades regularity for decay of the multiplier.
## Uniform $L^2$ Boundedness
The central analytic question is whether an order-zero semiclassical pseudodifferential operator is bounded on $L^2$ with constants independent of $h$. The answer is yes when finitely many symbol seminorms are uniformly controlled, and the result is the semiclassical Calderon-Vaillancourt theorem.
[quotetheorem:7305]
[citeproof:7305]
This theorem is the base estimate for the whole mapping theory. It says that the order-zero part of the calculus behaves like multiplication by a bounded function, provided finitely many symbol seminorms remain uniformly bounded. The uniform seminorm hypothesis is essential: a symbol such as $a_h(x,\xi)=b(x/h^2)\chi(\xi)$ has derivatives growing like powers of $h^{-1}$, so the theorem gives no uniform conclusion even though each fixed-$h$ operator is a legitimate pseudodifferential operator. The result also does not say that every bounded family of operators has an order-zero symbol, nor does it control symbols with oscillation outside the chosen semiclassical class. Its role is narrower and powerful: it turns symbolic order zero into uniform $L^2$ boundedness, which is the input for all Sobolev mapping estimates below.
[example: Compactly Supported Order Zero Symbol]
Let $a\in C_c^\infty(T^*\mathbb R^n)$ and assume that for every pair of multi-indices $\alpha,\beta$ there is a constant $M_{\alpha\beta}$ such that
\begin{align*}
\sup_{0<h\le h_0}\sup_{x,\xi}\left|\partial_x^\alpha\partial_\xi^\beta a(x,\xi;h)\right|\le M_{\alpha\beta}.
\end{align*}
Set $A_h=\operatorname{Op}_h(a)$. For the integer $N=N(n)$ appearing in the *Semiclassical Calderon Vaillancourt Theorem*, define
\begin{align*}
M_N=\sum_{|\alpha|+|\beta|\le N}\sup_{0<h\le h_0}\sup_{x,\xi}\left|\partial_x^\alpha\partial_\xi^\beta a(x,\xi;h)\right|.
\end{align*}
Each term in this finite sum is finite by the uniform derivative hypothesis, so $M_N<\infty$. Applying the *Semiclassical Calderon Vaillancourt Theorem* gives, for every $u\in\mathcal S(\mathbb R^n)$ and every $0<h\le h_0$,
\begin{align*}
\|A_hu\|_{L^2}\le C(n)M_N\|u\|_{L^2}.
\end{align*}
Since $\mathcal S(\mathbb R^n)$ is dense in $L^2(\mathbb R^n)$, $A_h$ extends to a bounded operator on $L^2(\mathbb R^n)$ with operator norm at most $C(n)M_N$, uniformly for $0<h\le h_0$. Thus a compactly supported phase-space cutoff whose finitely many relevant seminorms are uniformly controlled gives a reusable uniformly bounded microlocal cutoff.
[/example]
Uniform control of symbol seminorms is an actual hypothesis. If the symbol oscillates in $x$ or $\xi$ on a scale much smaller than the calculus permits, the theorem no longer gives a bounded family.
[example: Loss Of Uniformity From Rapid Symbol Oscillation]
Let $a_h(x,\xi)=b(x/h^2)\chi(\xi)$, where $b,\chi\in C_c^\infty(\mathbb R^n)$, $b$ is nonconstant, and $\chi$ is not identically zero. For a multi-index $\alpha$, the chain rule gives
\begin{align*}
\partial_x^\alpha a_h(x,\xi)=h^{-2|\alpha|}(\partial^\alpha b)(x/h^2)\chi(\xi).
\end{align*}
Taking suprema and using the change of variable $y=x/h^2$, we get
\begin{align*}
\sup_{x,\xi}|\partial_x^\alpha a_h(x,\xi)|=h^{-2|\alpha|}\sup_y|\partial^\alpha b(y)|\sup_\xi|\chi(\xi)|.
\end{align*}
Since $b$ is nonconstant, some first derivative $\partial_{x_j}b$ is not identically zero. For $\alpha=e_j$, this gives
\begin{align*}
\sup_{x,\xi}|\partial_{x_j}a_h(x,\xi)|=h^{-2}\sup_y|\partial_{x_j}b(y)|\sup_\xi|\chi(\xi)|.
\end{align*}
The two suprema on the right are fixed positive numbers, so this seminorm grows like $h^{-2}$ as $h\to 0$.
Thus the relevant order-zero symbol seminorms are not uniformly bounded in $h$. For each fixed $h>0$, the function $a_h$ is still a smooth compactly supported symbol, so the corresponding operator is controlled by fixed-$h$ estimates. What fails is the uniform hypothesis needed by the semiclassical order-zero calculus: the estimates obtained from symbol seminorms have constants that grow at least like a power of $h^{-1}$.
[/example]
## Order Reduction and Sobolev Mapping
Once $L^2$ boundedness is known for order-zero operators, the mapping theorem for order $m$ operators is a bookkeeping problem in the symbolic calculus. The problem is to measure the input in $H^s_h$, the output in $H^{s-m}_h$, and reduce the statement to an order-zero estimate.
[quotetheorem:7306]
[citeproof:7306]
The result encodes the meaning of symbol order: order $m$ costs $m$ semiclassical derivatives. The order hypothesis is necessary because the conjugated operator $\Lambda_h^{s-m}A_h\Lambda_h^{-s}$ is order zero only when the symbol of $A_h$ has order $m$; if the symbol grows faster in $\xi$, the same argument leaves a positive-order remainder and the target space is too regular. Uniform seminorm control is also part of the conclusion: parameter-dependent symbols with uncontrolled derivatives may satisfy fixed-$h$ estimates without giving constants stable as $h\to 0$. This theorem is therefore a precise bookkeeping principle, not a compactness or smoothing statement; it will be used whenever later arguments move pseudodifferential factors across semiclassical Sobolev weights.
[example: Elliptic Regularization By The Semiclassical Bessel Potential]
Let $u\in\mathcal S(\mathbb R^n)$ and set $A_h=\langle hD\rangle^{-s}=\operatorname{Op}_h(\langle \xi\rangle^{-s})$. On the ordinary Fourier side, the symbol variable is $\xi=h\zeta$, so
\begin{align*}
\widehat{A_hu}(\zeta)=\langle h\zeta\rangle^{-s}\widehat u(\zeta).
\end{align*}
Using the definition of the semiclassical Sobolev norm and then cancelling the multiplier pointwise gives
\begin{align*}
\|A_hu\|_{H^s_h}^2=\int_{\mathbb R^n}\langle h\zeta\rangle^{2s}\left|\langle h\zeta\rangle^{-s}\widehat u(\zeta)\right|^2\,d\zeta=\int_{\mathbb R^n}|\widehat u(\zeta)|^2\,d\zeta.
\end{align*}
By Plancherel's identity, the last integral is $\|u\|_{L^2}^2$. Therefore
\begin{align*}
\|A_hu\|_{H^s_h}=\|u\|_{L^2}
\end{align*}
for every $0<h\le h_0$, with constant exactly $1$.
Since $\mathcal S(\mathbb R^n)$ is dense in $L^2(\mathbb R^n)$, this identity extends $A_h$ uniquely to a bounded map $L^2(\mathbb R^n)\to H^s_h(\mathbb R^n)$ uniformly in $h$. The operator regularizes by $s$ semiclassical derivatives because the factor $\langle h\zeta\rangle^{-s}$ is exactly cancelled by the $H^s_h$ weight, so the semiclassical frequency scale is unchanged.
[/example]
The same theorem applies after inserting cutoffs in $x$ or $\xi$, and this is how local estimates are usually used. A local statement often begins by choosing $\chi,\psi\in C_c^\infty$ and estimating $\chi A_h\psi$ between semiclassical Sobolev spaces.
[remark: Dependence On Finitely Many Seminorms]
The mapping norm is controlled by finitely many derivatives of the symbol, not by the whole symbol class. The exact number is rarely important in qualitative microlocal arguments, but it matters in problems where symbols themselves depend on additional parameters besides $h$.
[/remark]
## Proper Support and Localization
Pseudodifferential operators on noncompact spaces have kernels that may interact with points far away in $x$. To make localization compatible with the calculus, one imposes proper support or inserts cutoffs so that compactly supported inputs produce compactly supported outputs.
[definition: Properly Supported Semiclassical Operator]
A semiclassical operator $A_h:C_c^\infty(\mathbb R^n)\to C^\infty(\mathbb R^n)$ with Schwartz kernel $K_A(x,y;h)$ is properly supported if the two coordinate projections
\begin{align*}
\operatorname{supp}K_A\to\mathbb R^n,\qquad (x,y)\mapsto x,\qquad
\operatorname{supp}K_A\to\mathbb R^n,\qquad (x,y)\mapsto y
\end{align*}
are proper maps.
[/definition]
Proper support is a support condition on the kernel, not a microlocal condition on the symbol. In local arguments one usually inserts cutoffs so that the tested input and output live in a compact coordinate region. The discarded pieces are harmless only after their supports have been separated enough to permit integration by parts; the next result records the precise Sobolev estimate used for those smoothing pieces throughout the calculus.
[quotetheorem:7308]
[citeproof:7308]
A smoothing remainder is invisible to semiclassical wavefront set calculations because it improves all Sobolev orders with rapid $h$ gain. The compact cutoffs in the theorem are necessary: a smooth kernel without support control can still have global growth or long-range interaction that prevents a uniform Sobolev estimate on the whole noncompact space. Rapid $h$-control is also essential, since ordinary smoothing improves regularity but may leave an $h$-sized contribution large enough to be detected by semiclassical wavefront tests. Thus identities modulo $h^\infty\Psi^{-\infty}$ are local, quantitative statements: after the correct cutoffs are inserted, the error is smaller than every power of $h$ in every Sobolev order. In the pseudolocality argument below, this is the step that turns separated microsupports into negligible terms: whenever two cutoffs live in disjoint phase-space neighborhoods, their composition is treated as one of these rapidly smoothing errors.
This prepares the qualitative mapping statement of the chapter. Once smoothing remainders have been separated from the analysis, the only possible source of a microlocal singularity is the phase-space region where the operator has non-negligible symbol. The next definition records that active region.
## Pseudolocality
The last mapping property is qualitative rather than a plain norm estimate. A pseudodifferential operator cannot create a singularity at a point of phase space where the input was microlocally smooth, except possibly where the operator itself has symbol support.
[definition: Semiclassical Microsupport]
Let $A_h=\operatorname{Op}_h(a):\mathcal S(\mathbb R^n)\to\mathcal S'(\mathbb R^n)$ be a semiclassical pseudodifferential operator. The semiclassical microsupport $\operatorname{MS}_h(A)$ is the closed subset of $T^*\mathbb R^n$ outside which the full symbol of $A_h$ is $O(h^\infty)$ in all symbol seminorms after localization.
[/definition]
The microsupport tells us where the operator is active in phase space. With this support notion available, the remaining problem is to prove that testing $A_hu_h$ near a phase-space point only depends on testing $u_h$ near the same point, up to smoothing errors from separated microsupports.
[quotetheorem:7309]
[proofunderconstruction:7309]
This theorem is the bridge from operator estimates to microlocal regularity. Proper support is needed so that the localized compositions in the proof act on distributions without uncontrolled behaviour at spatial infinity; without it, a kernel may import singular behaviour from far away into the region being tested. The $h$-tempered hypothesis is also necessary because wavefront tests measure rapid powers of $h$, and a family growing faster than every power of $h^{-1}$ can defeat the residual estimates used in the definition. Finally, the microsupport condition is the operator-side restriction: if the symbol is active near a point, pseudolocality says only that existing singularities are not moved off the diagonal in phase space, not that the operator is elliptically invertible there. The theorem justifies using pseudodifferential cutoffs as phase-space tests: if every cutoff near a point gives rapid Sobolev decay after being applied to $u_h$, then applying a pseudodifferential operator cannot introduce a new positive test at that point.
[example: Localization Away From The Symbol Support]
Let $A_h=\operatorname{Op}_h(a)$ and $B_h=\operatorname{Op}_h(b)$. The $\xi$-supports are separated: if $|\xi-\xi_0|<\varepsilon$, then $|\xi-\xi_0|$ cannot also be $>2\varepsilon$, so every point of $\operatorname{supp}a$ is outside $\operatorname{supp}b$.
The composition symbol $c\sim b\# a$ has asymptotic terms of the form
\begin{align*}
c_\alpha(x,\xi;h)=\frac{h^{|\alpha|}}{\alpha!}\partial_\xi^\alpha b(x,\xi;h)D_x^\alpha a(x,\xi;h).
\end{align*}
Differentiating a compactly supported function does not enlarge its support, so $\partial_\xi^\alpha b$ is still supported where $|\xi-\xi_0|>2\varepsilon$, while $D_x^\alpha a$ is still supported where $|\xi-\xi_0|<\varepsilon$. Hence
\begin{align*}
\partial_\xi^\alpha b(x,\xi;h)D_x^\alpha a(x,\xi;h)=0
\end{align*}
for every multi-index $\alpha$. Thus every coefficient in the symbolic expansion of $b\# a$ vanishes.
By the composition calculus with separated microsupports, the full composition therefore has symbol $O(h^N)$ in every $S^{-N}$ seminorm for every $N$. Equivalently,
\begin{align*}
B_hA_h\in h^\infty\Psi^{-\infty}_h.
\end{align*}
The smoothing remainder estimates then give, for all $s,t\in\mathbb R$ and every $N$,
\begin{align*}
\|B_hA_hu\|_{H^t_h}\le C_{s,t,N}h^N\|u\|_{H^s_h}.
\end{align*}
Thus a frequency cutoff supported away from the $\xi$-support of $a$ cannot detect $\operatorname{Op}_h(a)u$ except by a rapidly decaying $h^\infty$ error.
[/example]
The chapter's main estimates can now be summarized in one operational rule. Order-zero operators are uniformly $L^2$ bounded under uniform symbol control; order-$m$ operators map $H^s_h$ to $H^{s-m}_h$ by order reduction; properly supported operators localize correctly; and smoothing remainders are negligible between all semiclassical Sobolev spaces. These facts form the analytic foundation for the later definition of semiclassical wavefront sets and for elliptic regularity in phase space.
These Sobolev estimates give the right notion of size, but elliptic analysis requires more: one needs a symbolic inverse when the principal part does not vanish. The next chapter uses the calculus to build parametrices and show how ellipticity produces local inversion up to controlled remainders.
# 6. Ellipticity and Parametrices
Ellipticity is the microlocal condition that an operator has a symbolic inverse at the point of phase space under consideration. The previous chapters built the calculus needed to compose and localize semiclassical pseudodifferential operators; this chapter uses that calculus to turn nonvanishing of a principal symbol into an actual inverse modulo smoothing remainders. The central point is that ellipticity is not merely a global property of differential operators, but a local phase-space property that gives regularity and a priori estimates wherever the symbol does not vanish.
## Elliptic Symbols and Characteristic Sets
The first question is where a semiclassical operator can fail to be inverted. For a differential operator this obstruction is often described by the vanishing of its principal symbol, but in the semiclassical calculus the obstruction lives on $T^*X$ and must be measured uniformly as $h \to 0$.
[definition: Semiclassical Ellipticity on an Open Set]
Let $a \in S^m(T^*X)$ and let $U \subset T^*X$ be open. The symbol $a$ is semiclassically elliptic of order $m$ on $U$ if for every compact $K \subset U$ there exist constants $c_K>0$ and $h_K>0$ such that
\begin{align*}
|a(x,\xi;h)| \ge c_K \langle \xi \rangle^m
\end{align*}
for all $(x,\xi) \in K$ and all $0<h<h_K$.
[/definition]
This definition says that the symbol has the same size as the model order function on the region being studied, not merely at fiber infinity. If $a(x_0,\xi_0;h)$ vanishes to leading order, no lower bound of this kind can hold on a small compact neighbourhood of $(x_0,\xi_0)$, and the multiplication model $u \mapsto a(x_0,\xi_0)u$ already shows why an inverse should fail there. In applications the cutoff is often concentrated near one covector rather than spread over a named open set. We therefore need a pointwise formulation that can be invoked whenever a single phase-space point is known to lie away from the bad region.
[definition: Ellipticity at a Phase-Space Point]
Let $a \in S^m(T^*X)$ and let $(x_0,\xi_0) \in T^*X$. The symbol $a$ is elliptic at $(x_0,\xi_0)$ if there is an open neighbourhood $U$ of $(x_0,\xi_0)$ such that $a$ is semiclassically elliptic of order $m$ on $U$.
[/definition]
At finite covectors this condition is often expressed by the nonvanishing of the principal symbol. For estimates it is not enough to test one point at a time; we also need a geometric set that records all possible obstructions. This leads to the characteristic set, the phase-space locus where the principal symbol vanishes.
[definition: Characteristic Set]
Let $A=\operatorname{Op}_h(a) \in \Psi_h^m(X)$ be regarded as an operator
\begin{align*}
A:C_c^\infty(X)\longrightarrow \mathcal D'(X),
\end{align*}
and, by semiclassical Sobolev mapping, as a continuous map $A:H_h^s(X)\to H_h^{s-m}(X)$ for each $s\in\mathbb R$. Let $\sigma_h(A)$ denote its principal symbol in the order-$m$ semiclassical symbol quotient. The characteristic set of $A$ is
\begin{align*}
\operatorname{Char}_h(A)=\{(x,\xi)\in T^*X : \sigma_h(A)(x,\xi)=0\}
\end{align*}
inside the region where the principal symbol is defined.
[/definition]
The complement of the characteristic set is the elliptic region, subject to the order estimates just described. Microlocal analysis studies estimates after cutting away from $\operatorname{Char}_h(A)$.
[example: Shifted Semiclassical Laplacian]
On $\mathbb R^n$, take
\begin{align*}
P=h^2\Delta+1
\end{align*}
with $\Delta=-\sum_{i=1}^n\partial_{x_i}^2$. Since $hD_{x_i}=-ih\partial_{x_i}$ has semiclassical symbol $\xi_i$, we have
\begin{align*}
h^2\Delta=-h^2\sum_{i=1}^n\partial_{x_i}^2=\sum_{i=1}^n(hD_{x_i})^2
\end{align*}
and therefore the semiclassical principal symbol is
\begin{align*}
p(x,\xi)=\sum_{i=1}^n\xi_i^2+1=|\xi|^2+1.
\end{align*}
With $\langle \xi\rangle=(1+|\xi|^2)^{1/2}$, this is exactly
\begin{align*}
p(x,\xi)=\langle \xi\rangle^2.
\end{align*}
Thus $|p(x,\xi)|=\langle \xi\rangle^2$ on all of $T^*\mathbb R^n$, so the elliptic lower bound holds with constant $c_K=1$ on every compact $K$. Hence $P$ is elliptic of order $2$ everywhere, and the leading symbol of its inverse is
\begin{align*}
p(x,\xi)^{-1}=(|\xi|^2+1)^{-1}.
\end{align*}
This model has no characteristic points because $|\xi|^2+1$ never vanishes.
[/example]
This model has no characteristic set, which is why it is the cleanest place to see symbolic inversion. More interesting operators are elliptic only away from an energy surface.
[example: Schrödinger Elliptic Region]
Let $P=h^2\Delta+V(x)-E$ on an open set $X\subset \mathbb R^n$, where $V\in C^\infty(X;\mathbb R)$, $E\in\mathbb R$, and $\Delta=-\sum_{i=1}^n\partial_{x_i}^2$. Since $hD_{x_i}=-ih\partial_{x_i}$ has semiclassical symbol $\xi_i$, we have
\begin{align*}
h^2\Delta=-h^2\sum_{i=1}^n\partial_{x_i}^2=\sum_{i=1}^n(hD_{x_i})^2.
\end{align*}
Therefore the order-$2$ semiclassical principal symbol of $P$ is
\begin{align*}
p(x,\xi)=\sum_{i=1}^n\xi_i^2+V(x)-E=|\xi|^2+V(x)-E.
\end{align*}
Now let $K\subset \{(x,\xi):|\xi|^2+V(x)\ne E\}$ be compact. The function $|p|$ is continuous and is strictly positive on $K$, so it has a positive minimum
\begin{align*}
\delta_K=\min_{(x,\xi)\in K}|p(x,\xi)|>0.
\end{align*}
Also $\langle \xi\rangle^2=1+|\xi|^2$ is continuous on $K$, so
\begin{align*}
M_K=\max_{(x,\xi)\in K}\langle \xi\rangle^2<\infty.
\end{align*}
For every $(x,\xi)\in K$,
\begin{align*}
|p(x,\xi)|\ge \delta_K=\frac{\delta_K}{M_K}M_K\ge \frac{\delta_K}{M_K}\langle \xi\rangle^2.
\end{align*}
Thus the elliptic lower bound holds on every compact subset away from the energy surface $p^{-1}(0)=\{|\xi|^2+V(x)=E\}$, with $c_K=\delta_K/M_K$. Hence $P$ is microlocally elliptic precisely away from that energy surface.
[/example]
The second example is the standard bridge to propagation theory: elliptic regularity controls the solution away from the classical energy surface, while the later transport results study what happens along it.
## Parametrices by Symbolic Inversion
Once a symbol is elliptic, the next problem is to construct an inverse inside the pseudodifferential calculus. Since composition is not pointwise multiplication but the semiclassical star product, the inverse must be built as an asymptotic series whose coefficients correct the lower-order errors.
There are two useful versions of this construction. The first is a Euclidean high-frequency parametrix: the symbol is assumed invertible for large $|\xi|$, so the inverse removes high-frequency components modulo ordinary smoothing. This is not yet the fully microlocal theorem, but it is the clean model for symbolic inversion at fiber infinity.
[quotetheorem:7307]
[citeproof:7307]
The high-frequency hypothesis is stronger and more global than local ellipticity. It requires a uniform lower bound outside a fixed fiber ball, so the inverse symbol can be constructed in the exterior region $|\xi|\ge R$. The conclusion is correspondingly global in frequency but weaker near the low-frequency ball, where the theorem only promises an ordinary smoothing remainder. This is exactly the right model for elliptic regularity at fiber infinity, but microlocal analysis also needs inverses on arbitrary open regions of $T^*X$. The following definition packages that local version.
[definition: Semiclassical Parametrix]
Let $A\in \Psi_h^m(X)$ be regarded as a map $A:C_c^\infty(X)\to \mathcal D'(X)$ and as a continuous map $A:H_h^s(X)\to H_h^{s-m}(X)$ for every $s\in\mathbb R$. Let $U\subset T^*X$ be open. A semiclassical parametrix for $A$ on $U$ is an operator $B\in \Psi_h^{-m}(X)$, regarded as $B:C_c^\infty(X)\to \mathcal D'(X)$ and $B:H_h^s(X)\to H_h^{s+m}(X)$, such that
\begin{align*}
BA=I+R_L, \qquad AB=I+R_R
\end{align*}
microlocally on $U$, where $R_L,R_R\in h^\infty\Psi_h^{-\infty}(X)$ are residual operators $C_c^\infty(X)\to C^\infty(X)$ microlocally on $U$.
[/definition]
The remainders are negligible in the semiclassical calculus: after localization in $U$, they improve regularity by arbitrary order and decay faster than any power of $h$. The definition leaves open the main construction problem: when does an elliptic symbol actually admit such a microlocal inverse? The answer is the parametrix theorem, obtained by correcting the naive inverse $a_m^{-1}$ order by order in the symbolic product.
[quotetheorem:7310]
[citeproof:7310]
The theorem is the main payoff of the symbolic calculus. It turns the algebraic statement $a_m\ne0$ into an operator-level inverse, but only after accepting that the inverse is microlocal and modulo negligible remainders.
The ellipticity hypothesis is essential. If $A=hD_x$ on $\mathbb R$ and we microlocalize near $(x_0,0)$, then the principal symbol $\xi$ vanishes there; an inverse would have to divide by $\xi$, which is not a symbol on any neighbourhood meeting $\xi=0$. The microlocal qualifier is also essential: an operator may be elliptic on one open subset of $T^*X$ and characteristic elsewhere, so the construction cannot produce a global inverse from local nonvanishing. Finally, the smoothing remainder is the natural endpoint of the asymptotic construction; the symbolic recursion kills errors to every power of $h$, but an infinite symbolic expansion need not converge to an exact inverse.
[example: First Terms for the Laplacian Parametrix]
For $P=h^2\Delta+1$ on $\mathbb R^n$, with $\Delta=-\sum_{i=1}^n\partial_{x_i}^2$, the semiclassical symbol of $hD_{x_i}=-ih\partial_{x_i}$ is $\xi_i$. Hence
\begin{align*}
h^2\Delta=-h^2\sum_{i=1}^n\partial_{x_i}^2=\sum_{i=1}^n(hD_{x_i})^2,
\end{align*}
so the full symbol of $P$ is
\begin{align*}
p(\xi)=\sum_{i=1}^n\xi_i^2+1=|\xi|^2+1.
\end{align*}
Set
\begin{align*}
b(\xi)=(|\xi|^2+1)^{-1}.
\end{align*}
Because both $b$ and $p$ are independent of $x$, every term in the semiclassical composition formula involving an $x$-derivative vanishes. Thus the star product reduces to pointwise multiplication:
\begin{align*}
(b\# p)(\xi)=b(\xi)p(\xi).
\end{align*}
Substituting the two symbols gives
\begin{align*}
b(\xi)p(\xi)=(|\xi|^2+1)^{-1}(|\xi|^2+1)=1.
\end{align*}
Similarly,
\begin{align*}
(p\# b)(\xi)=p(\xi)b(\xi)=(|\xi|^2+1)(|\xi|^2+1)^{-1}=1.
\end{align*}
Therefore $\operatorname{Op}_h(b)P=I$ and $P\operatorname{Op}_h(b)=I$ in this constant-coefficient model. The parametrix is exact, so the symbolic expansion stops at its leading term and no lower-order correction symbols are needed.
[/example]
When the coefficients vary in $x$, the same first term remains visible, but derivatives of the coefficients generate the subsequent correction terms. The recursive construction records precisely how noncommutativity of $x$ and $hD_x$ affects inversion.
[remark: Left and Right Parametrices]
For scalar elliptic symbols the left and right parametrices have the same principal symbol. Their lower-order coefficients may be produced by different recursive equations, but the difference is absorbed by a smoothing semiclassical remainder in the microlocal region where both inverses are constructed.
[/remark]
This distinction matters for operators on vector bundles and systems, where matrix-valued symbols must be inverted and left-right order may interact with noncommuting matrix coefficients.
## Microlocal Elliptic Estimates
The parametrix becomes analytically useful only after it is paired with Sobolev mapping estimates. The guiding question is: if $Au$ is controlled near a phase-space point where $A$ is elliptic, how much control of $u$ follows there?
[definition: Semiclassical Sobolev Norm]
For $s\in\mathbb R$, let
\begin{align*}
\operatorname{Op}_h(\langle \xi\rangle^s):\mathcal S'(\mathbb R^n)\longrightarrow \mathcal S'(\mathbb R^n)
\end{align*}
be the semiclassical Fourier multiplier with symbol $\langle \xi\rangle^s$. The semiclassical Sobolev space is
\begin{align*}
H_h^s(\mathbb R^n)=\{u\in\mathcal S'(\mathbb R^n):\operatorname{Op}_h(\langle \xi\rangle^s)u\in L^2(\mathbb R^n)\},
\end{align*}
and the semiclassical Sobolev norm is the functional $\|\cdot\|_{H_h^s}:H_h^s(\mathbb R^n)\to[0,\infty)$ defined by
\begin{align*}
\|u\|_{H_h^s} = \|\operatorname{Op}_h(\langle \xi\rangle^s)u\|_{L^2}.
\end{align*}
[/definition]
The factor $h$ in the quantization means that derivatives are measured as powers of $hD_x$, so the resulting estimates are uniform as $h\to0$. With this scale of Sobolev spaces fixed, the parametrix identity should become a norm inequality. The next theorem is the localized estimate obtained by cutting off $u$, cutting off $Au$, and absorbing the remaining term into a rapidly decaying smoothing error.
[quotetheorem:7311]
[citeproof:7311]
The estimate is local in both $x$ and $\xi$. It says that failure of regularity of $u$ at an elliptic point can only come from failure of regularity of $Au$ at the same point, up to remainders invisible in the semiclassical wavefront set.
Each hypothesis rules out a real failure mode. Without ellipticity, $A=hD_x$ near $\xi=0$ cannot control the low-frequency part of $u$, since constants are annihilated by $hD_x$. Without the microsupport restriction on $C$, the cutoff may see characteristic points where the parametrix was never constructed. The compact support, or in later formulations a proper support and polynomial boundedness assumption, keeps the residual term meaningful in Sobolev spaces. The negative Sobolev remainder is not cosmetic: a residual operator is very smoothing, but it still acts on the distribution $u$, so some coarse global control is needed before its $h^N$ gain can be used.
[example: Localizing Near One Phase-Space Point]
Fix $(x_0,\xi_0)\in T^*\mathbb R^n$ and let $A=\operatorname{Op}_h(a)\in\Psi_h^0$ with $a(x_0,\xi_0)\ne0$. Since $a$ is continuous, there is a ball $B$ around $(x_0,\xi_0)$ such that
\begin{align*}
|a(x,\xi)|\ge c_0>0
\end{align*}
for all $(x,\xi)\in B$. Choose $c\in C_c^\infty(T^*\mathbb R^n)$ with $\operatorname{supp}c\Subset B$ and $c=1$ on a smaller neighbourhood of $(x_0,\xi_0)$, and set $C=\operatorname{Op}_h(c)$. On $\operatorname{supp}c$, the order-$0$ elliptic lower bound is
\begin{align*}
|a(x,\xi)|\ge c_0=c_0\langle \xi\rangle^0.
\end{align*}
Applying the *Microlocal Elliptic Estimate* with $m=0$ gives a cutoff $C_1\in\Psi_h^0$ elliptic on $\operatorname{WF}_h(C)$ such that, for every $N$, there is a constant $C_{s,N}$ with
\begin{align*}
\|Cu\|_{H_h^s}\le C_{s,N}\|C_1Au\|_{H_h^s}+C_{s,N}h^N\|u\|_{H_h^{-N}}.
\end{align*}
The Sobolev order on $Au$ is $s$ rather than $s-0$, because $A$ has order $0$. Thus, after cutting to the region where $a$ is nonvanishing, control of $Au$ in $H_h^s$ controls $u$ in $H_h^s$, up to a residual term that is negligible when $u$ has a coarse negative Sobolev bound.
[/example]
This localization is why ellipticity at a single point is a useful concept rather than a technical refinement. It lets the calculus separate regular and singular regions before any global boundary condition or global inverse is introduced.
## Elliptic Regularity for Semiclassical Pseudodifferential Operators
The final question is how the microlocal estimate translates into regularity statements. If $Au$ has semiclassical Sobolev regularity and $A$ is elliptic on a region, then the parametrix transfers that regularity back to $u$ on the same region, with the expected gain of $m$ derivatives.
[definition: Semiclassical Wavefront Regularity]
Let $u=(u_h)_{0<h<h_0}$ be an $h$-dependent family in $\mathcal D'(X)$. We say that $u$ is $H_h^s$ microlocally near $(x_0,\xi_0)$ if there exists $C\in\Psi_h^0(X)$, regarded as a continuous operator $C:\mathcal D'(X)\to\mathcal D'(X)$ and $C:H_h^s(X)\to H_h^s(X)$, elliptic at $(x_0,\xi_0)$ such that $Cu=O(1)$ in $H_h^s$ as $h\to0$.
[/definition]
This notion records regularity after applying a phase-space cutoff. Its failure defines the corresponding Sobolev wavefront obstruction, so the natural regularity question is whether ellipticity can remove that obstruction. The next theorem answers this by transferring $H_h^{s-m}$ control of $Au$ into $H_h^s$ control of $u$ on the elliptic region.
[quotetheorem:7312]
[citeproof:7312]
This theorem is the microlocal version of classical elliptic regularity. The ellipticity assumption cannot be removed: for $A=hD_x$, the equation $Au=0$ is compatible with nonzero constant families, so control of $Au$ near $\xi=0$ gives no positive derivative control of $u$ there. The polynomial lower Sobolev bound is also needed to make the residual term harmless; without some coarse control of $u$, the factor $h^N$ may be defeated by a family growing faster than every power of $h^{-1}$.
The conclusion is deliberately local. It does not say that $u$ is globally regular, does not solve boundary conditions, and does not describe what happens on $\operatorname{Char}_h(A)$. Instead it removes the elliptic region from the singular analysis, leaving later propagation, spectral, and Fredholm arguments to study the characteristic set and any global solvability conditions.
[example: Elliptic Regularity Away from an Energy Surface]
For $P=h^2\Delta+V(x)-E$ with $\Delta=-\sum_{i=1}^n\partial_{x_i}^2$, the semiclassical symbol is
\begin{align*}
p(x,\xi)=|\xi|^2+V(x)-E.
\end{align*}
Let $K\subset T^*X$ be compact and suppose
\begin{align*}
K\cap\{(x,\xi):|\xi|^2+V(x)=E\}=\varnothing.
\end{align*}
Then $p$ is continuous and nonzero on $K$, so
\begin{align*}
\delta_K=\min_{(x,\xi)\in K}|p(x,\xi)|>0.
\end{align*}
Since $\langle\xi\rangle^2=1+|\xi|^2$ is continuous on $K$,
\begin{align*}
M_K=\max_{(x,\xi)\in K}\langle\xi\rangle^2<\infty.
\end{align*}
For every $(x,\xi)\in K$,
\begin{align*}
|p(x,\xi)|\ge \delta_K.
\end{align*}
Also $\langle\xi\rangle^2\le M_K$, hence
\begin{align*}
\frac{\delta_K}{M_K}\langle\xi\rangle^2\le \delta_K.
\end{align*}
Combining these two inequalities gives
\begin{align*}
|p(x,\xi)|\ge \frac{\delta_K}{M_K}\langle\xi\rangle^2.
\end{align*}
Thus $P$ is elliptic of order $2$ on a neighbourhood of $K$.
Choose $C\in\Psi_h^0$ with $\operatorname{WF}_h(C)\Subset K$, and suppose $Pu=f$. By the *Microlocal Elliptic Estimate* with $m=2$, there is a cutoff $C_1\in\Psi_h^0$ elliptic on $\operatorname{WF}_h(C)$ such that, for every $N$,
\begin{align*}
\|Cu\|_{H_h^s}\le C_N\|C_1Pu\|_{H_h^{s-2}}+C_Nh^N\|u\|_{H_h^{-N}}.
\end{align*}
Substituting $Pu=f$ gives
\begin{align*}
\|Cu\|_{H_h^s}\le C_N\|C_1f\|_{H_h^{s-2}}+C_Nh^N\|u\|_{H_h^{-N}}.
\end{align*}
If $f$ is bounded in $H_h^{s-2}$ microlocally near $K$, then $\|C_1f\|_{H_h^{s-2}}=O(1)$. Therefore $Cu$ is bounded in $H_h^s$ modulo the residual term $C_Nh^N\|u\|_{H_h^{-N}}$, which is negligible whenever $u$ has the corresponding coarse negative Sobolev control. Hence semiclassical singularities of $u$ over $K$ cannot occur unless they are already forced by the right-hand side $f$ or by failure of the residual bound.
[/example]
The energy surface is therefore the only place where this operator can carry new microlocal information. Later propagation results will study how singularities move along the Hamilton flow on that surface, while the elliptic theory developed here removes all points away from it.
[remark: Global Versus Microlocal Inverses]
A microlocal parametrix need not solve the global equation $Au=f$ or respect boundary conditions. It is an inverse only after phase-space localization, and the remainder is negligible only in that localized sense. Global Fredholm theory and boundary value problems require additional compactness, boundary, or radiation hypotheses.
[/remark]
Parametrices reveal where an operator can be inverted microlocally, which naturally leads to the question of where a family fails to be negligible. The next chapter answers that by defining semiclassical wavefront sets, using the operators already constructed to track singularities in phase space.
# 7. Semiclassical Wavefront Sets
Semiclassical wavefront sets answer a local phase-space question: where does a family $u_h$ fail to be negligible after cutting both in position and in frequency at the $h^{-1}$ scale? Earlier chapters built the symbol classes and the operators $A=\operatorname{Op}_h(a)$ that test such localisation. The prerequisites for this chapter are the semiclassical symbol classes, the quantization map $\operatorname{Op}_h$, residual operator estimates, ellipticity, and the microlocal parametrix construction from the pseudodifferential calculus. This chapter turns that calculus into a definition of microlocal support for $h$-dependent distributions, then compares the operator definition with concrete Fourier tests in Euclidean charts.
The guiding principle is that a point $(x_0,\xi_0)\in T^*X$ is absent from $\operatorname{WF}_h(u_h)$ when some order-zero semiclassical pseudodifferential operator, elliptic near that point, kills $u_h$ to infinite order in $h$. The same language also records behaviour at fiber infinity, where $|\xi|$ is large in the compactified cotangent fibers. By the end of the chapter, wavefront sets will behave like support sets under the semiclassical calculus: operators cannot create microlocal mass away from their own microsupport, and elliptic annihilators detect absence of wavefront.
## Testing Phase Space by Annihilators
The first problem is to turn the heuristic phrase "no $h$-frequency near $(x_0,\xi_0)$" into a coordinate-invariant condition. Position cutoffs alone only see ordinary support, while Fourier cutoffs alone ignore where the oscillation occurs. Semiclassical pseudodifferential operators combine both tests in a way stable under changes of charts.
[definition: Semiclassical Tempered Family]
Let $X$ be a smooth manifold. A family $u_h\in \mathcal{D}'(X)$, $0<h\le h_0$, is semiclassically tempered if for every compact $K\subset X$ there exist $N\ge 0$ and a continuous seminorm $p_K$ on $C_c^\infty(X)$ such that
\begin{align*}
|u_h(\phi)|\le C_K h^{-N} p_K(\phi)
\end{align*}
for all $\phi\in C_c^\infty(K)$ and all $0<h\le h_0$.
[/definition]
Temperedness supplies the growth condition under which pseudodifferential remainders act predictably: residual operators turn tempered inputs into $O(h^\infty)$ outputs. Once this polynomial control is in place, the next task is to decide which individual phase-space points can be removed by an elliptic local test. That leads to the finite-point definition of $\operatorname{WF}_h(u_h)$.
[definition: Semiclassical Wavefront Set at Finite Points]
Let $u_h$ be a semiclassically tempered family on $X$. A point $q\in T^*X$ is not in $\operatorname{WF}_h(u_h)$ if there exists an operator $A\in \Psi_h^0(X)$, acting continuously $A:C_c^\infty(X)\to C^\infty(X)$ and by duality on $\mathcal D'(X)$ after the usual proper-support localization, such that $A$ is elliptic at $q$ and
\begin{align*}
Au_h=O(h^\infty)_{C^\infty}
\end{align*}
microlocally near the base projection of $q$. The semiclassical wavefront set $\operatorname{WF}_h(u_h)$ is the complement of the set of such points in $T^*X$.
[/definition]
Here $O(h^\infty)_{C^\infty}$ means that every local $C^k$ seminorm is $O(h^N)$ for every $N$. Since the annihilator must be elliptic at $q$, it is a genuine test of $u_h$ near $q$ rather than an operator whose symbol vanishes at the point being tested.
[example: Pure Oscillation with Compact Amplitude]
Let $u_h(x)=e^{ix\cdot \xi_0/h}a(x)$ on $\mathbb R^n$, with $a\in C_c^\infty(\mathbb R^n)$ and $\xi_0\in\mathbb R^n$. We show that
\begin{align*}
\operatorname{WF}_h(u_h)=\operatorname{supp}(a)\times\{\xi_0\}.
\end{align*}
If $x_1\notin \operatorname{supp}(a)$, then some $\chi\in C_c^\infty(\mathbb R^n)$ satisfies $\chi(x_1)\ne0$ and $\chi a=0$, so $\chi u_h=0$; hence no point over $x_1$ lies in the wavefront set. If $\xi_1\ne\xi_0$, choose $b(x,\xi)$ supported where $|\xi-\xi_0|\ge \delta>0$ near $(x_1,\xi_1)$. For the standard left quantization,
\begin{align*}
\operatorname{Op}_h(b)u_h(x)=(2\pi h)^{-n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\eta/h}b(x,\eta)e^{iy\cdot\xi_0/h}a(y)\,dy\,d\eta.
\end{align*}
The phase is $x\cdot\eta+y\cdot(\xi_0-\eta)$, so
\begin{align*}
\nabla_y\bigl(x\cdot\eta+y\cdot(\xi_0-\eta)\bigr)=\xi_0-\eta.
\end{align*}
On the support of $b$, $|\xi_0-\eta|\ge\delta$, so repeated integration by parts in $y$ with the operator
\begin{align*}
L=\frac{h}{i|\xi_0-\eta|^2}(\xi_0-\eta)\cdot\nabla_y
\end{align*}
gives $Le^{iy\cdot(\xi_0-\eta)/h}=e^{iy\cdot(\xi_0-\eta)/h}$, and each integration by parts gains one factor of $h$ while differentiating only the compactly supported smooth function $a$. Thus $\operatorname{Op}_h(b)u_h=O(h^\infty)_{C^\infty}$ near every covector different from $\xi_0$.
It remains to see what happens over $\xi_0$. Substituting $\eta=\xi_0+\zeta$ in the same formula gives
\begin{align*}
\operatorname{Op}_h(b)u_h(x)=e^{ix\cdot\xi_0/h}(2\pi h)^{-n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\zeta/h}b(x,\xi_0+\zeta)a(y)\,dy\,d\zeta.
\end{align*}
The inner oscillatory operator has principal term $b(x,\xi_0)a(x)$, so
\begin{align*}
\operatorname{Op}_h(b)u_h(x)=e^{ix\cdot\xi_0/h}b(x,\xi_0)a(x)+O(h)_{C^\infty_{\mathrm{loc}}}.
\end{align*}
If $x_1\in\operatorname{supp}(a)$ and $b$ is elliptic at $(x_1,\xi_0)$, then $b(x,\xi_0)$ is nonzero on a neighbourhood of $x_1$, and every such neighbourhood contains points where $a$ is nonzero. The leading factor $b(x,\xi_0)a(x)$ therefore cannot vanish identically near $x_1$, so the output is not $O(h^\infty)$ there. Thus the only remaining points are exactly the support of the amplitude over the single semiclassical covector $\xi_0$.
[/example]
This example shows the role of $hD_x$: the ordinary function $u_h$ is smooth for every fixed $h$, but its derivatives grow like powers of $h^{-1}$. The relevant obstruction is therefore not pointwise differentiability, but lack of uniform rapid decay after a test tuned to the phase $\xi_0$. This is why semiclassical wavefront sets are useful even for smooth oscillatory families: they remember where the family carries $h$-scale oscillation, and not merely where a fixed distribution is singular. The wavefront set records this moving high-frequency information rather than classical singularities.
[remark: Ordinary Smoothness Does Not Remove Semiclassical Wavefront]
A family may consist entirely of smooth functions and still have a nonempty semiclassical wavefront set. The obstruction is not lack of differentiability at fixed $h$, but failure of uniform $O(h^\infty)$ decay after a phase-space cutoff.
[/remark]
The annihilator definition is local in the finite part of phase space, so it should not depend on the chosen elliptic test there. However, finite covectors alone do not yet control all possible failure of rapid decay. A family such as $e^{ix/h^2}\chi(x)$ has no bounded semiclassical frequency near any fixed finite covector, but its ordinary frequency escapes beyond the scale recorded by $T^*X$. The global criterion for rapid decay therefore has to wait until the wavefront set has been extended to fiber infinity.
## Fourier Tests in Euclidean Charts
The operator definition is invariant, but computations usually happen in coordinates. The central question is how to recognise absence from $\operatorname{WF}_h(u_h)$ using a local cutoff and the semiclassical Fourier transform.
[definition: Semiclassical Fourier Transform]
For $0<h\le h_0$, the semiclassical Fourier transform is the map $\mathcal F_h:\mathcal S(\mathbb R^n)\to\mathcal S(\mathbb R^n)$ defined by
\begin{align*}
\mathcal F_h v_h(\xi)=(2\pi h)^{-n/2}\int_{\mathbb R^n}e^{-ix\cdot \xi/h}v_h(x)\,d\mathcal L^n(x).
\end{align*}
[/definition]
The variable $\xi$ is the covector variable, not the ordinary Fourier frequency. Ordinary Fourier concentration near $\xi/h$ becomes semiclassical Fourier concentration near $\xi$. To use this transform for wavefront sets, we must combine it with a spatial cutoff and then compare rapid decay near a covector with elliptic annihilation by pseudodifferential operators.
[quotetheorem:7313]
[citeproof:7313]
This theorem is the main computational bridge in the chapter. It says that the pseudodifferential definition is not hiding a new notion of frequency; it is the invariant form of local semiclassical Fourier decay. The hypotheses are local and coordinate-dependent in appearance: $U\subset\mathbb R^n$ is needed so that the formula for $\mathcal F_h$ has a fixed meaning, while the invariant operator definition is what lets the same test survive a change of chart. Without the spatial cutoff $\chi$, a Fourier transform can detect the covector $\xi_0$ but cannot say where in the base variable the oscillation occurs: if $u_h=e^{ix/h}\chi_1(x)+e^{ix/h}\chi_2(x)$ with disjoint compactly supported cutoffs, the uncut transform sees the same covector $1$ from both packets and cannot distinguish the two base supports. Without the neighbourhood $V$ of $\xi_0$, decay at isolated covectors would miss nearby frequency leakage, for instance $e^{ix(1+h^{1/2})/h}\chi(x)$ approaches covector $1$ while never being exactly at $\xi=1$ for fixed $h$. Without temperedness, the converse can fail at the remainder stage: multiplying a fixed compactly supported smooth function by $e^{1/h}$ gives a non-tempered family for which a residual operator with kernel $e^{-1/(2h)}k(x,y)$ may produce a term of size $e^{1/(2h)}$, which is not $O(h^\infty)$.
[example: Gaussian Packet at One Phase-Space Point]
Let
\begin{align*}
\nu_h(x)=h^{-n/4}e^{-|x-x_0|^2/(2h)}e^{i(x-x_0)\cdot \xi_0/h}
\end{align*}
on $\mathbb R^n$. Its $L^2$ norm is independent of $h$, since
\begin{align*}
\|\nu_h\|_{L^2}^2=h^{-n/2}\int_{\mathbb R^n}e^{-|x-x_0|^2/h}\,dx
\end{align*}
and the change of variables $z=(x-x_0)/h^{1/2}$ gives
\begin{align*}
\|\nu_h\|_{L^2}^2=\int_{\mathbb R^n}e^{-|z|^2}\,dz=\pi^{n/2}.
\end{align*}
The semiclassical Fourier transform is explicitly concentrated at $\xi_0$. Indeed,
\begin{align*}
\mathcal F_h\nu_h(\xi)=(2\pi h)^{-n/2}h^{-n/4}\int_{\mathbb R^n}e^{-ix\cdot \xi/h}e^{-|x-x_0|^2/(2h)}e^{i(x-x_0)\cdot \xi_0/h}\,dx.
\end{align*}
With $x=x_0+h^{1/2}z$, this becomes
\begin{align*}
\mathcal F_h\nu_h(\xi)=(2\pi)^{-n/2}h^{-n/4}e^{-ix_0\cdot\xi/h}\int_{\mathbb R^n}e^{-|z|^2/2}e^{-iz\cdot(\xi-\xi_0)/h^{1/2}}\,dz.
\end{align*}
The standard Gaussian integral
\begin{align*}
\int_{\mathbb R^n}e^{-|z|^2/2}e^{-iz\cdot\eta}\,dz=(2\pi)^{n/2}e^{-|\eta|^2/2}
\end{align*}
with $\eta=(\xi-\xi_0)/h^{1/2}$ gives
\begin{align*}
\mathcal F_h\nu_h(\xi)=h^{-n/4}e^{-ix_0\cdot\xi/h}e^{-|\xi-\xi_0|^2/(2h)}.
\end{align*}
If a cutoff $\chi$ is supported where $|x-x_0|\ge \varepsilon$, then each $x$-derivative of $\chi\nu_h$ is a finite sum of terms bounded by $C h^{-M}e^{-\varepsilon^2/(2h)}$, hence is $O(h^N)$ for every $N$ because exponential decay beats every power of $h$. Therefore no point with base coordinate $x\ne x_0$ lies in $\operatorname{WF}_h(\nu_h)$. If $\chi=1$ near $x_0$, then $(1-\chi)\nu_h=O(h^\infty)_{C^\infty}$ by the same estimate, so $\mathcal F_h(\chi\nu_h)$ differs from $\mathcal F_h\nu_h$ by an $O(h^\infty)$ Schwartz function. On any neighbourhood with $|\xi-\xi_0|\ge \varepsilon$, the displayed formula gives
\begin{align*}
|\mathcal F_h\nu_h(\xi)|\le h^{-n/4}e^{-\varepsilon^2/(2h)}=O(h^\infty),
\end{align*}
with the same conclusion after multiplying by any power of $\langle\xi\rangle$. Thus no point with covector $\xi\ne\xi_0$ lies in the wavefront set.
It remains to check that $(x_0,\xi_0)$ is actually present. Let $\chi\in C_c^\infty(\mathbb R^n)$ satisfy $\chi(x_0)\ne0$. At $\xi=\xi_0$,
\begin{align*}
\mathcal F_h(\chi\nu_h)(\xi_0)=(2\pi h)^{-n/2}h^{-n/4}e^{-ix_0\cdot\xi_0/h}\int_{\mathbb R^n}\chi(x)e^{-|x-x_0|^2/(2h)}\,dx.
\end{align*}
Changing variables $x=x_0+h^{1/2}z$ gives
\begin{align*}
\mathcal F_h(\chi\nu_h)(\xi_0)=(2\pi)^{-n/2}h^{-n/4}e^{-ix_0\cdot\xi_0/h}\int_{\mathbb R^n}\chi(x_0+h^{1/2}z)e^{-|z|^2/2}\,dz.
\end{align*}
By dominated convergence, the integral tends to
\begin{align*}
\chi(x_0)\int_{\mathbb R^n}e^{-|z|^2/2}\,dz=\chi(x_0)(2\pi)^{n/2},
\end{align*}
so $|\mathcal F_h(\chi\nu_h)(\xi_0)|$ is bounded below by a positive multiple of $h^{-n/4}$ for all sufficiently small $h$. Hence the local Fourier decay condition fails at $(x_0,\xi_0)$, and
\begin{align*}
\operatorname{WF}_h(\nu_h)=\{(x_0,\xi_0)\}.
\end{align*}
The packet is therefore spread over an $h^{1/2}$ scale in both position and semiclassical frequency, but its microlocal support collapses to the single phase-space point $(x_0,\xi_0)$.
[/example]
The Gaussian packet illustrates why the wavefront set is a closed subset of phase space rather than only a frequency support. Both the base point and the covector are needed to locate where oscillation and concentration occur simultaneously.
[example: Semiclassical Oscillation Without Classical Singular Support]
Let $u_h(x)=e^{ix/h}\chi(x)$ on $\mathbb R$, with $\chi\in C_c^\infty(\mathbb R)$ nonzero. For each fixed $h>0$, the factor $x\mapsto e^{ix/h}$ is smooth and $\chi$ is smooth, so $u_h\in C_c^\infty(\mathbb R)$; hence the classical singular support of $u_h$ is empty.
Now fix $x_1$ with $\chi(x_1)\ne 0$. If $\psi\in C_c^\infty(\mathbb R)$ satisfies $\psi(x_1)\ne0$, then $\psi(x_1)\chi(x_1)\ne0$ after replacing $\psi$ by a smaller cutoff if necessary. At the semiclassical covector $\xi=1$,
\begin{align*}
\mathcal F_h(\psi u_h)(1)=(2\pi h)^{-1/2}\int_{\mathbb R}e^{-ix/h}e^{ix/h}\psi(x)\chi(x)\,dx.
\end{align*}
Since $e^{-ix/h}e^{ix/h}=1$, this is
\begin{align*}
\mathcal F_h(\psi u_h)(1)=(2\pi h)^{-1/2}\int_{\mathbb R}\psi(x)\chi(x)\,dx.
\end{align*}
Choose $\psi$ supported so close to $x_1$ that $\psi\chi$ has constant nonzero sign, or constant nonzero complex argument, on the support of $\psi$. Then
\begin{align*}
\int_{\mathbb R}\psi(x)\chi(x)\,dx\ne0.
\end{align*}
Therefore $|\mathcal F_h(\psi u_h)(1)|=c\,h^{-1/2}$ for some $c>0$, so the localized semiclassical Fourier transform is not $O(h^\infty)$ near $\xi=1$. Thus $(x_1,1)\in\operatorname{WF}_h(u_h)$ for every $x_1$ with $\chi(x_1)\ne0$. The family is classically smooth at each fixed $h$, but it carries persistent $h$-scale oscillation at covector $1$.
[/example]
The Fourier criterion is also useful for comparing different coordinate charts. On overlaps, the phase transform induced by a diffeomorphism sends covectors by the cotangent lift, and stationary phase transfers rapid decay away from the transformed covector.
[remark: Coordinate Invariance of the Fourier Test]
If $y=\kappa(x)$ is a diffeomorphism between coordinate charts, then covectors transform by $\eta=(D\kappa_x^{-1})^\top\xi$. The Fourier-local definition in the $x$ chart and the corresponding definition in the $y$ chart define the same subset of $T^*X$ because the oscillatory kernel for pullback has a unique stationary point at the cotangent-transformed covector.
[/remark]
## Fiber Infinity and Microsupport
Finite covectors detect oscillation with bounded semiclassical frequency $\xi$. The next issue is what happens when frequencies escape to $|\xi|=\infty$, because order-zero operators also have symbols with behaviour at the boundary of compactified fibers.
[definition: Fiber-Radial Compactification of the Cotangent Bundle]
Let $X$ be a smooth manifold. The fiber-radial compactification $\overline{T}^*X$ is obtained by compactifying each cotangent fiber $T_x^*X$ to a closed ball whose boundary records directions $\hat\xi=\xi/|\xi|$ as $|\xi|\to\infty$. The boundary $S^*X=\partial\overline{T}^*X$ is the cosphere bundle.
[/definition]
The boundary points of $\overline{T}^*X$ are not additional base points; they are directions of unbounded covectors over ordinary points of $X$. Semiclassical wavefront sets should therefore be closed subsets of $\overline{T}^*X$, with a finite part and a fiber-infinity part. The next definition extends elliptic annihilation to this boundary by using conic neighbourhoods of directions.
[definition: Semiclassical Wavefront Set at Fiber Infinity]
Let $u_h$ be semiclassically tempered on $X$. A point $q\in S^*X$ is not in $\operatorname{WF}_h(u_h)$ if there exists an operator $A\in\Psi_h^0(X)$, acting continuously $A:C_c^\infty(X)\to C^\infty(X)$ and by duality on $\mathcal D'(X)$ after the usual proper-support localization, whose principal symbol is elliptic in a conic neighbourhood of $q$ at fiber infinity and such that
\begin{align*}
Au_h=O(h^\infty)_{C^\infty}
\end{align*}
microlocally near the base projection of $q$.
[/definition]
Together with the finite-point definition, this gives a closed subset of the fiber-radial compactification $\overline{T}^*X$. From now on $\operatorname{WF}_h(u_h)$ denotes this compactified semiclassical wavefront set: its interior part lies in $T^*X$, and its boundary part lies in $S^*X$.
At fiber infinity this definition resembles the classical wavefront condition, but the estimates remain uniform in $h$. It detects high ordinary frequencies beyond the bounded semiclassical scale. A classical wavefront set records directions in which a fixed distribution is not smooth; the fiber-infinity part of $\operatorname{WF}_h$ records directions in which an $h$-dependent smooth family has ordinary frequencies too large to remain in a bounded semiclassical window. With the compactified object now defined, the elliptic-parametrix argument gives the promised global regularity criterion.
[quotetheorem:7314]
[citeproof:7314]
The compactification is essential in this statement. On a compact base, absence of the finite part alone still allows frequencies to escape to $|\xi|=\infty$; for example, $u_h(x)=e^{ix/h^2}\chi(x)$ on a coordinate patch has no bounded finite semiclassical covector, but it is not rapidly small where $\chi$ is nonzero. On noncompact manifolds an additional base-compactness issue appears: a family $u_h(x)=\phi(x-h^{-1})$ on $\mathbb R$ is rapidly zero on every fixed compact set for small $h$, yet it is not $O(h^\infty)$ in any global $C^0(\mathbb R)$ norm. Temperedness is also part of the mechanism: residual operators have $O(h^\infty)$ kernels, and the polynomial growth assumption prevents applying such a remainder to $u_h$ from losing rapid decay.
Since operators can themselves be localised or negligible in selected regions of $\overline{T}^*X$, we also need a matching support notion for pseudodifferential operators. The wavefront set belongs to the input family, while an operator has its own phase-space footprint determined by where its full symbol is not residual. This distinction matters because an operator may be active only in a narrow frequency band, or only over a small region of the base manifold, even when its Schwartz kernel is globally defined. The next definition isolates the exact region in which an operator can affect microlocal information, so that the later pseudolocality statement can separate limitations coming from the input from limitations coming from the operator itself.
[definition: Semiclassical Microsupport of an Operator]
Let $A\in\Psi_h^m(X)$ be an operator acting continuously $A:C_c^\infty(X)\to C^\infty(X)$ and by duality on $\mathcal D'(X)$ after the usual proper-support localization. In a local coordinate chart, let $a:U\times\mathbb R^n\times(0,h_0]\to\mathbb C$ be a full symbol for $A$, written $a(x,\xi;h)$. The semiclassical microsupport $\operatorname{MS}_h(A)$ is the complement in $\overline{T}^*X$ of the set of points $q$ for which $a$ is $O(h^\infty\langle\xi\rangle^{-\infty})$ in a neighbourhood of $q$, with the corresponding conic interpretation at fiber infinity.
[/definition]
Microsupport is the phase-space support of an operator rather than of a distribution. It is the set where the operator can interact with a wavefront set. The natural compatibility question is whether applying $A$ can create wavefront outside the region where $A$ acts or outside the wavefront already present in $u_h$.
[quotetheorem:7309]
[citeproof:7309]
Pseudolocality says that semiclassical pseudodifferential operators do not create new singular directions; they can only filter what is already present at the same phase-space point. The temperedness hypothesis is used exactly where residual remainders appear: an $O(h^\infty)$ smoothing operator applied to a family growing faster than every power of $h^{-1}$ need not produce an $O(h^\infty)$ output. A concrete failure is obtained by taking $u_h=e^{1/h}\chi$ with $\chi\in C_c^\infty(X)$ nonzero and applying a residual operator $R_h$ whose kernel is $e^{-1/(2h)}k(x,y)$ for a fixed smoothing kernel $k$; then $R_hu_h$ has size comparable to $e^{1/(2h)}$ on points where the smoothed function is nonzero, so the output is not rapidly small. The inclusion also has a built-in limitation: if $A$ is not elliptic at $q$, then $A$ may erase wavefront there, so the result gives no way to recover $q\in\operatorname{WF}_h(u_h)$ from $Au_h$. For elliptic operators, filtering is reversible microlocally because a parametrix recovers the input modulo residual errors. This motivates the sharper inclusion-and-equivalence statement used throughout elliptic microlocal arguments.
[quotetheorem:7315]
[proofunderconstruction:7315]
The elliptic part of this result is the operational content of wavefront testing: elliptic operators preserve exactly the microlocal information at points where they are elliptic. Non-elliptic operators may erase information, but the inclusion prevents them from inventing it. The failure mode is concrete: a frequency cutoff whose symbol vanishes near the active covector of an oscillatory family can remove that covector from the output, so no implication from $Au_h$ back to $u_h$ is available there. Elliptic regularity arguments use the opposite situation: when the symbol is invertible at $q$, the parametrix turns information about $Au_h$ into information about $u_h$ at the same point.
[example: Frequency Cutoff Removing a Pure Oscillation]
Let $u_h(x)=e^{ix\cdot \xi_0/h}a(x)$ on $\mathbb R^n$, with $a\in C_c^\infty(\mathbb R^n)$. For the left semiclassical quantization,
\begin{align*}\operatorname{Op}_h(b)u_h(x)=(2\pi h)^{-n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\eta/h}b(x,\eta)e^{iy\cdot\xi_0/h}a(y)\,dy\,d\eta.\end{align*}
Combining the exponentials gives
\begin{align*}e^{i(x-y)\cdot\eta/h}e^{iy\cdot\xi_0/h}=e^{ix\cdot\eta/h}e^{iy\cdot(\xi_0-\eta)/h}.\end{align*}
Assume first that $b$ is supported, over the support of $a$, where $|\eta-\xi_0|\ge \delta>0$. On this support,
\begin{align*}L=\frac{h}{i|\xi_0-\eta|^2}(\xi_0-\eta)\cdot\nabla_y\end{align*}
satisfies
\begin{align*}Le^{iy\cdot(\xi_0-\eta)/h}=e^{iy\cdot(\xi_0-\eta)/h}.\end{align*}
Integrating by parts $N$ times in $y$ therefore moves $(L^*)^N$ onto $a(y)$ and gives a factor $h^N|\xi_0-\eta|^{-N}$ times derivatives of $a$. Since $a$ is smooth and compactly supported, all those differentiated amplitudes are bounded on a fixed compact set. Thus every local $C^k$ seminorm of $\operatorname{Op}_h(b)u_h$ is $O(h^N)$ for every $N$, so $Au_h=O(h^\infty)_{C^\infty}$.
Now suppose instead that $b(x_1,\xi_0)a(x_1)\ne0$ at some point $x_1$. Put $\eta=\xi_0+\zeta$. Then
\begin{align*}\operatorname{Op}_h(b)u_h(x)=e^{ix\cdot\xi_0/h}(2\pi h)^{-n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\zeta/h}b(x,\xi_0+\zeta)a(y)\,dy\,d\zeta.\end{align*}
The term with $\zeta=0$ in the amplitude gives the identity kernel:
\begin{align*}(2\pi h)^{-n}\int_{\mathbb R^n}\int_{\mathbb R^n}e^{i(x-y)\cdot\zeta/h}b(x,\xi_0)a(y)\,dy\,d\zeta=b(x,\xi_0)a(x).\end{align*}
Writing
\begin{align*}b(x,\xi_0+\zeta)-b(x,\xi_0)=\sum_{j=1}^n \zeta_j\int_0^1\partial_{\xi_j}b(x,\xi_0+t\zeta)\,dt\end{align*}
shows that the remaining oscillatory integral has one factor $\zeta_j$. Since
\begin{align*}\zeta_j e^{i(x-y)\cdot\zeta/h}=\frac{h}{i}\partial_{x_j}e^{i(x-y)\cdot\zeta/h},\end{align*}
that remainder is $h$ times another order-zero semiclassical operator applied to $a$, hence is $O(h)_{C^\infty_{\mathrm{loc}}}$. Therefore
\begin{align*}\operatorname{Op}_h(b)u_h(x)=e^{ix\cdot\xi_0/h}b(x,\xi_0)a(x)+O(h)_{C^\infty_{\mathrm{loc}}}.\end{align*}
Near $x_1$, the smooth function $b(x,\xi_0)a(x)$ is not identically zero, so the output cannot be $O(h^\infty)$ there. The cutoff removes the oscillation only when its symbol misses the covector $\xi_0$ over the support of $a$; when the symbol is nonzero at that covector, the same semiclassical wavefront remains.
[/example]
These results complete the basic microlocal dictionary for this first course. The wavefront set is defined by elliptic annihilation, computed by local semiclassical Fourier decay, extended to compactified fibers, and controlled by microsupport under the pseudodifferential calculus. The same dictionary is the entry point for several later subjects: propagation of singularities studies how Hamiltonian flow transports $\operatorname{WF}_h$ for evolution equations; spectral asymptotics use compactified phase-space localisation to count eigenfunctions; scattering theory uses the fiber-infinity part to separate incoming and outgoing behaviour; and quantum mechanics reads $h$ as Planck's constant, so a semiclassical wavefront set records the classical phase-space states that survive in the high-frequency limit.
Wavefront sets identify the surviving phase-space content of a family, but one still needs positivity to turn symbolic information into operator inequalities. The next chapter develops sharp G\r{}arding estimates, showing how nonnegative symbols yield lower bounds that are strong enough for spectral and energy arguments.
# 8. Positivity and Sharp Garding Estimates
The symbolic calculus built earlier gives precise control of products and commutators, but it leaves a basic analytic question unresolved: when does a real symbol give an operator bounded below on $L^2$? In elementary spectral theory a nonnegative multiplication operator has a nonnegative quadratic form, while a nonnegative Fourier multiplier does the same after Plancherel. A general pseudodifferential operator mixes position and frequency, and quantization introduces oscillatory averaging, so pointwise positivity of the symbol is not inherited exactly. This chapter explains the replacement principle used throughout microlocal energy estimates: nonnegative symbols give lower bounds with losses controlled by powers of $h$.
## The Positivity Problem for Quantization
The first difficulty is to separate positivity of a function on phase space from positivity of the operator obtained by quantizing it. For a real symbol $a(x,\xi)$, the expression $(\operatorname{Op}_h(a)u,u)_{L^2}$ should be thought of as measuring the phase-space average of $a$ against $u$, but the quantization map is not an order-preserving functional calculus. The obstruction is already visible in the difference between multiplying by $a(x)$, applying a Fourier multiplier $a(\xi)$, and quantizing a symbol depending on both variables.
[definition: Semiclassical Lower Bound]
Let $D\subset L^2(\mathbb R^n)$ be a dense subspace containing $\mathcal S(\mathbb R^n)$, and let
\begin{align*}
A_h:D\to L^2(\mathbb R^n)
\end{align*}
be a family of symmetric operators on $D$. We say that $A_h$ has semiclassical lower bound $-C h^\alpha$ on $L^2$ if there exist $C>0$ and $h_0>0$ such that
\begin{align*}
(A_hu,u)_{L^2} \ge -C h^\alpha \|u\|_{L^2}^2
\end{align*}
for every $u\in D$ and $0<h\le h_0$.
[/definition]
This definition records the kind of positivity that survives quantization. The exponent $\alpha$ depends on the quantization and on the symbolic hypotheses; sharp Garding gives an $O(h)$ loss for nonnegative order-zero Weyl symbols, while Fefferman-Phong improves the scale for certain second-order lower bounds.
[example: Positive Multiplication and Positive Fourier Multiplier]
Let $u\in \mathcal S(\mathbb R^n)$ and let $a\in C_b^\infty(\mathbb R^n)$ satisfy $a(x)\ge 0$ for every $x$. For the multiplication operator $M_a u=a u$, the $L^2$ inner product gives
\begin{align*}
(M_a u,u)_{L^2}=\int_{\mathbb R^n}a(x)u(x)\overline{u(x)}\,d\mathcal L^n(x).
\end{align*}
Since $u(x)\overline{u(x)}=|u(x)|^2$ and $a(x)\ge 0$, each pointwise integrand is nonnegative:
\begin{align*}
a(x)u(x)\overline{u(x)}=a(x)|u(x)|^2\ge 0.
\end{align*}
Therefore
\begin{align*}
(M_a u,u)_{L^2}=\int_{\mathbb R^n}a(x)|u(x)|^2\,d\mathcal L^n(x)\ge 0.
\end{align*}
Now let $b\in C_b^\infty(\mathbb R^n)$ satisfy $b(\xi)\ge 0$ for every $\xi$, and write $\widehat u_h=\mathcal F_hu$. The semiclassical Fourier multiplier is defined by $b(hD)=\mathcal F_h^{-1}M_b\mathcal F_h$, where $M_bv(\xi)=b(\xi)v(\xi)$. Using the unitarity of $\mathcal F_h$ on $L^2$,
\begin{align*}
(b(hD)u,u)_{L^2}=(M_b\widehat u_h,\widehat u_h)_{L^2_\xi}.
\end{align*}
Expanding the $L^2_\xi$ inner product gives
\begin{align*}
(M_b\widehat u_h,\widehat u_h)_{L^2_\xi}=\int_{\mathbb R^n}b(\xi)\widehat u_h(\xi)\overline{\widehat u_h(\xi)}\,d\mathcal L^n(\xi).
\end{align*}
Since $\widehat u_h(\xi)\overline{\widehat u_h(\xi)}=|\mathcal F_hu(\xi)|^2$ and $b(\xi)\ge 0$,
\begin{align*}
(b(hD)u,u)_{L^2}=\int_{\mathbb R^n}b(\xi)|\mathcal F_hu(\xi)|^2\,d\mathcal L^n(\xi)\ge 0.
\end{align*}
Thus multiplication by a nonnegative function is positive in physical space, while a nonnegative Fourier multiplier is positive after passing to the semiclassical Fourier representation.
[/example]
The example explains why positivity is tempting: in the two separated cases, nonnegativity of the symbol is an exact operator statement. The mixed case has no representation that diagonalises all symbols, so the best general result is a lower bound whose error is small as $h\to 0$.
[remark: Quantization Is Not Order Preserving]
The implication $a\ge 0\implies \operatorname{Op}_h(a)\ge 0$ fails for standard quantizations when $a$ depends on both $x$ and $\xi$. The failure is not a defect of a poor convention; it reflects the noncommutativity of $x_j$ and $hD_{x_j}$. Weyl quantization is preferred for positivity estimates because it treats $x$ and $\xi$ symmetrically and gives real symmetric operators from real symbols.
[/remark]
## Weyl Quantization and Symbolic Square Roots
The next problem is to manufacture positivity indirectly. If $a\ge 0$ and $a=b^2$, then one would like to write $\operatorname{Op}_h^w(a)$ as $(\operatorname{Op}_h^w(b))^*\operatorname{Op}_h^w(b)$ plus lower-order terms. The symbolic product formula makes this possible, but the square root $b=a^{1/2}$ may not be a classical symbol near zeros of $a$, so ellipticity or regularisation hypotheses matter.
[definition: Positive Elliptic Symbol]
Let $m:T^*\mathbb R^n\to (0,\infty)$ be an order function. A real symbol $a\in S(m)$ is positive elliptic if there exist constants $c>0$ and $R>0$ such that
\begin{align*}
a(x,\xi)\ge c\,m(x,\xi)
\end{align*}
whenever $|(x,\xi)|\ge R$.
[/definition]
Positive ellipticity gives enough room to take a symbolic square root away from compact sets and then patch the low-frequency region by adding a harmless positive constant. The next theorem makes this strategy precise, turning an elliptic lower bound into a positive square modulo a symbolic error that can be estimated by the Weyl calculus.
[quotetheorem:7316]
[citeproof:7316]
The uniform lower bound $a\ge cm$ is the point that makes the symbolic square root harmless: it keeps derivatives of $a^{1/2}$ from becoming singular. If $a(x,\xi)=x_1^2$ near $x_1=0$, then $a^{1/2}=|x_1|$ is not smooth across the zero set, so the square-root construction cannot be used in the same symbol class. The theorem therefore does not say that every nonnegative symbol is a positive Weyl square, nor does it give an exact positivity statement for quantization. Its role is elliptic: when the symbol is bounded away from zero at the scale measured by $m$, positivity can be recovered by factorisation up to a controlled symbolic error.
[example: Harmonic Oscillator Symbol]
For $a(x,\xi)=|\xi|^2+|x|^2=\sum_{j=1}^n(\xi_j^2+x_j^2)$ on $T^*\mathbb R^n$, the terms depending only on $\xi$ quantize as Fourier multipliers and the terms depending only on $x$ quantize as multiplication operators, so
\begin{align*}
\operatorname{Op}_h^w(a)=\sum_{j=1}^n\left((hD_{x_j})^2+x_j^2\right).
\end{align*}
Set
\begin{align*}
A_j=hD_{x_j}-ix_j.
\end{align*}
On $\mathcal S(\mathbb R^n)$, $hD_{x_j}$ is symmetric and multiplication by $ix_j$ has adjoint multiplication by $-ix_j$, hence
\begin{align*}
A_j^*=hD_{x_j}+ix_j.
\end{align*}
Using $D_{x_j}=-i\partial_{x_j}$, the commutator with multiplication by $x_j$ is
\begin{align*}
[hD_{x_j},x_j]u=hD_{x_j}(x_ju)-x_jhD_{x_j}u=-ih\,u.
\end{align*}
Therefore
\begin{align*}
A_j^*A_j=(hD_{x_j}+ix_j)(hD_{x_j}-ix_j)=(hD_{x_j})^2+x_j^2-i[hD_{x_j},x_j].
\end{align*}
Substituting $[hD_{x_j},x_j]=-ih$ gives
\begin{align*}
A_j^*A_j=(hD_{x_j})^2+x_j^2-h.
\end{align*}
Equivalently,
\begin{align*}
(hD_{x_j})^2+x_j^2=A_j^*A_j+h.
\end{align*}
Summing over $j$ gives
\begin{align*}
\operatorname{Op}_h^w(a)=\sum_{j=1}^nA_j^*A_j+nh.
\end{align*}
Taking the $L^2$ inner product with $u\in\mathcal S(\mathbb R^n)$,
\begin{align*}
(\operatorname{Op}_h^w(a)u,u)_{L^2}=\sum_{j=1}^n\|A_ju\|_{L^2}^2+nh\|u\|_{L^2}^2.
\end{align*}
Since each norm square is nonnegative,
\begin{align*}
(\operatorname{Op}_h^w(a)u,u)_{L^2}\ge nh\|u\|_{L^2}^2.
\end{align*}
Thus this quadratic symbol is not merely nonnegative after quantization: the creation-annihilation factorisation exhibits a positive spectral scale of order $h$.
[/example]
This example is special because the quadratic symbol has an exact algebraic structure. For compactly supported nonnegative symbols or symbols vanishing to finite order, the same conclusion requires a general estimate rather than a direct factorisation.
## Sharp Garding Inequality
The central question is now: if $a\in S^0$ is real and nonnegative everywhere, how far below zero can $\operatorname{Op}_h^w(a)$ fall? The answer is sharp at the $h$ scale. This is the estimate that converts microlocal sign information into coercive energy inequalities with a controlled error.
[quotetheorem:7317]
[citeproof:7317]
The word sharp refers to the size of the loss. Exact positivity fails because Weyl quantization is not an order-preserving map: even for compactly supported $a\ge 0$, interference between neighbouring phase-space packets can contribute a negative quadratic-form error. The real-valued hypothesis is needed so that $\operatorname{Op}_h^w(a)$ is symmetric; without it, a lower bound for the real quadratic form is not the right statement. The nonnegativity hypothesis cannot be weakened to a sign-changing symbol, since testing on wave packets centred where $a<0$ detects a negative leading contribution. The order-zero assumption is also part of the estimate: it keeps the operator uniformly $L^2$-bounded, and higher-order symbols require a different scale such as the Fefferman-Phong bound below. Thus sharp Garding gives a lower bound with an $O(h)$ loss, not positivity and not a spectral gap.
[example: Compactly Supported Nonnegative Symbol]
Let $a\in C_c^\infty(T^*\mathbb R^n)$, assume $a(x,\xi)\ge 0$ for every $(x,\xi)$, and let $u\in \mathcal S(\mathbb R^n)$. Since $a$ is smooth with compact support, for every pair of multiindices $\alpha,\beta$ the derivative $\partial_x^\alpha\partial_\xi^\beta a$ is bounded. Thus $a$ is an order-zero symbol, so *Sharp Garding Inequality* applies to $a$ and gives constants $C>0$ and $h_0>0$ such that
\begin{align*}
(\operatorname{Op}_h^w(a)u,u)_{L^2}\ge -Ch\|u\|_{L^2}^2
\end{align*}
for every $0<h\le h_0$.
This conclusion does not come from elliptic factorisation. If $K=\operatorname{supp} a$, then $a(x,\xi)=0$ for $(x,\xi)\notin K$; hence no estimate of the form $a(x,\xi)\ge c\,m(x,\xi)$ can hold outside a compact set for an order function $m>0$ and a constant $c>0$. The hypothesis that $a$ vanishes somewhere also means that the pointwise square root need not satisfy symbol estimates in the same way an elliptic square root does; for instance, near a simple quadratic zero one meets the model $\sqrt{x_1^2}=|x_1|$, which is not smooth at $x_1=0$. Thus sharp Garding is the appropriate positivity tool near characteristic sets: it converts the degenerate sign condition $a\ge 0$ into the operator lower bound above, with the controlled loss $Ch\|u\|_{L^2}^2$.
[/example]
The compactly supported example is the microlocal model: cutoffs localise the analysis to a bounded region of phase space, and the operator may lose exact positivity through the cutoff itself. Sharp Garding says that this loss is lower order in the semiclassical parameter.
[remark: Dependence on Quantization]
For left or right Kohn-Nirenberg quantization, the corresponding positivity statement is usually reduced to Weyl quantization by changing symbols. The change of quantization modifies the symbol by terms of size $h$ in lower symbolic order. Thus the same $O(h)$ lower bound persists after replacing the symbol by its Weyl representative.
[/remark]
## Fefferman-Phong Type Lower Bounds
Sharp Garding is often enough for first-order energy estimates, but second-order operators require a more refined lower bound. The relevant question is whether a nonnegative symbol of order two can have a lower bound with an $h^2$ loss rather than the $h$ loss suggested by applying order-zero estimates after rescaling.
[quotetheorem:7318]
[proofunderconstruction:7318]
This result is especially useful when the principal energy is second order, for instance in semiclassical Schrodinger operators. The order-two hypothesis is not cosmetic: for $a(x,\xi)=|\xi|^2+V(x)$ with $V\ge 0$, the natural operator scale is $h^2D^2+V$, so an $h^2$ loss matches the differential order of the energy. Nonnegativity is still essential; if $a$ is negative at one phase-space point, coherent states centred there give a negative leading contribution that no $Ch^2$ remainder can hide. The theorem also does not say that $\operatorname{Op}_h^w(a)$ is nonnegative or that the same $h^2$ loss holds for arbitrary higher-order symbol classes without the correct metric and seminorm assumptions.
[example: Naive Positivity versus Weyl Positivity up to Error]
Let $a$ be real-valued, nonnegative, and depend on both $x$ and $\xi$. If Weyl quantization preserved order, then for every $u\in\mathcal S(\mathbb R^n)$ one would have
\begin{align*}
(\operatorname{Op}_h^w(a)u,u)_{L^2}\ge 0.
\end{align*}
That implication is false in general for mixed symbols, so the usable statement for $a\in S^0$ is the lower bound supplied by *Sharp Garding Inequality*: there are constants $C>0$ and $h_0>0$ such that, for $0<h\le h_0$,
\begin{align*}
(\operatorname{Op}_h^w(a)u,u)_{L^2}\ge -Ch\|u\|_{L^2}^2.
\end{align*}
Equivalently, the possible negative part of the quadratic form is controlled by an order-$h$ multiple of the $L^2$ mass:
\begin{align*}
(\operatorname{Op}_h^w(a)u,u)_{L^2}+Ch\|u\|_{L^2}^2\ge 0.
\end{align*}
If the same nonnegativity hypothesis is placed on an order-two symbol in the Fefferman-Phong class, *Basic Fefferman-Phong Inequality* gives constants $C>0$ and $h_0>0$ such that
\begin{align*}
(\operatorname{Op}_h^w(a)u,u)_{L^2}\ge -Ch^2\|u\|_{L^2}^2
\end{align*}
for every $u\in\mathcal S(\mathbb R^n)$ and $0<h\le h_0$. Since $h^2$ is smaller than $h$ for $0<h\le 1$, the order-two estimate allows a smaller negative error at the semiclassical scale. Thus the sign condition $a\ge 0$ is not enough by itself; the symbolic order determines which power of $h$ appears in the operator lower bound.
[/example]
The example is a bookkeeping warning. Positivity estimates are not merely qualitative; the power of $h$ determines whether an error can be absorbed into the main energy inequality.
## Role in Microlocal Energy Estimates
The final step is to connect these lower bounds to the energy method used later in propagation and elliptic regularity. Microlocal estimates often begin with an identity involving a commutator or a localized operator whose principal symbol has a sign. Sharp Garding turns that symbolic sign into an operator inequality, at the cost of a lower-order term.
[definition: Microlocal Positive Commutator Term]
Let $D\subset L^2(\mathbb R^n)$ be a common dense invariant domain containing $\mathcal S(\mathbb R^n)$. Let $P_h:D\to L^2(\mathbb R^n)$ be a semiclassical pseudodifferential operator, and let $A_h=\operatorname{Op}_h^w(a)$ with $a\in S^0$ real, initially acting on $D$. A microlocal positive commutator term is an operator $D\to L^2(\mathbb R^n)$ of the form
\begin{align*}
\frac{i}{h}[P_h^*,A_h^*A_h]-\operatorname{Op}_h^w(b)
\end{align*}
where the commutator is evaluated on $D$ and the principal symbol $b$ is real and nonnegative on the phase-space region under consideration.
[/definition]
After the commutator calculation has produced a nonnegative symbol, the analytic part is supplied by sharp Garding. The localized theorem below is needed because energy estimates see positivity only after multiplication by a cutoff, so the lower bound must be stated on the microlocal region where the symbol has a sign.
[quotetheorem:7319]
[citeproof:7319]
This theorem is the operational form of the chapter. The cutoff $\chi$ is needed because a commutator calculation usually produces a sign only on the microlocal region where the solution has been localized; outside that region, $b$ may have no useful sign. If $b<0$ at a point of $\operatorname{supp}\chi$, wave packets concentrated near that point produce a negative leading contribution, so Garding cannot convert the term into a harmless lower-order error. The theorem also controls only the localized quadratic form $(B_hu,u)_{L^2}$; it does not estimate the uncut operator, remove remainders from the commutator identity, or replace the separate elliptic and propagation estimates needed elsewhere.
[example: Localized Energy Absorption]
Suppose an energy identity contains
\begin{align*}
\|A_hu\|_{L^2}^2+(\operatorname{Op}_h^w(\chi^2b)u,u)_{L^2}\le R_h(u).
\end{align*}
Assume $b\ge 0$ on $\operatorname{supp}\chi$, and set
\begin{align*}
B_h=\operatorname{Op}_h^w(\chi^2b).
\end{align*}
By *Garding Step in a Microlocal Energy Estimate*, there are constants $C>0$ and $h_0>0$ such that, for $0<h\le h_0$,
\begin{align*}
(B_hu,u)_{L^2}\ge -Ch\|u\|_{L^2}^2.
\end{align*}
Adding $\|A_hu\|_{L^2}^2$ to both sides gives
\begin{align*}
\|A_hu\|_{L^2}^2+(B_hu,u)_{L^2}\ge \|A_hu\|_{L^2}^2-Ch\|u\|_{L^2}^2.
\end{align*}
Combining this lower bound with the assumed energy identity gives
\begin{align*}
\|A_hu\|_{L^2}^2-Ch\|u\|_{L^2}^2\le R_h(u).
\end{align*}
Adding $Ch\|u\|_{L^2}^2$ to both sides yields
\begin{align*}
\|A_hu\|_{L^2}^2\le R_h(u)+Ch\|u\|_{L^2}^2.
\end{align*}
Thus the nonnegative localized symbol contributes no uncontrolled negative term; its only effect is the semiclassical error $Ch\|u\|_{L^2}^2$, which must be absorbed together with the remaining terms in $R_h(u)$.
[/example]
The chapter's main lesson is that positivity in semiclassical analysis is stable but not exact. Weyl quantization, symbolic square roots, sharp Garding, and Fefferman-Phong estimates form a hierarchy: exact factorisation when available, $O(h)$ lower bounds for nonnegative order-zero symbols, and sharper $O(h^2)$ bounds for nonnegative second-order symbols. These estimates are the bridge from phase-space geometry to operator inequalities.
The positivity theory is now in place in Euclidean coordinates, but many applications live on curved spaces. The next chapter transports the calculus to manifolds, replacing the global Fourier transform with coordinate patches, partitions of unity, and invariant principal symbols on $T^*M$.
# 9. Pseudodifferential Operators on Manifolds
The Euclidean calculus developed earlier is local in the base variable: it quantizes symbols on open subsets of $\mathbb R^n$ and proves estimates by Fourier analysis in coordinates. On a manifold there is no global Fourier transform in the base variable, so the main problem is to identify which parts of the calculus survive changes of chart. This chapter explains how semiclassical pseudodifferential operators are assembled from local coordinate patches, why the leading symbol is an invariant function on $T^*M$, and how choices such as partitions of unity and half-density conventions affect only lower-order data.
## Coordinate Changes and Principal Symbols on the Cotangent Bundle
The first question is what a local symbol should become when the base coordinates are changed. A chart change acts not only on $x$ but also on covectors, and this cotangent transformation is exactly what makes the leading term coordinate-independent.
Let $M$ be a smooth $n$-manifold and let $(U,\kappa)$ be a coordinate chart, with $\kappa:U\to \kappa(U)\subset \mathbb R^n$. A function $u$ on $U$ is represented in coordinates by $u_\kappa=u\circ \kappa^{-1}$, and local semiclassical quantization applies to $u_\kappa$.
[definition: Cotangent Lift of a Chart]
Let $(U,\kappa)$ be a chart on a smooth manifold $M$. The induced cotangent coordinate map is the map
\begin{align*}
T^*U \longrightarrow \kappa(U)\times \mathbb R^n
\end{align*}
given by
\begin{align*}
(x,\xi) \longmapsto \bigl(\kappa(x),(d\kappa_x^{-1})^*\xi\bigr).
\end{align*}
[/definition]
This definition records how a covector is pulled into coordinate variables. If $y=\kappa(x)$, then a covector $\eta$ in the $y$-coordinates corresponds to the covector $\xi=(d\kappa_x)^*\eta$ on $M$. A simple reparametrisation shows why the fibre coordinate itself cannot be the invariant object.
[example: Cotangent Coordinates Under a Reparametrisation]
Let $M=\mathbb R$ and use the reparametrisation $y=\kappa(x)=x^3$ on $(0,\infty)$. For a phase function $\phi$, the covector $d\phi$ can be written in the $x$-coordinate as $\xi\,dx$ and in the $y$-coordinate as $\eta\,dy$, so
\begin{align*}
\xi\,dx=\eta\,dy.
\end{align*}
Differentiate $y=x^3$ with respect to $x$:
\begin{align*}
dy=d(x^3)=3x^2\,dx.
\end{align*}
Substituting this into $\xi\,dx=\eta\,dy$ gives
\begin{align*}
\xi\,dx=\eta(3x^2\,dx)=3x^2\eta\,dx.
\end{align*}
Since $dx$ is the coordinate covector basis, equality of the coefficients gives
\begin{align*}
\xi=3x^2\eta.
\end{align*}
Thus the fibre coordinate changes by the factor $3x^2$, while the covector $d\phi$ itself is unchanged as a geometric object.
[/example]
This example explains why a principal symbol must be viewed on $T^*M$ rather than as a formula depending on a particular coordinate momentum variable. The obstruction to a global calculus is that two coordinate formulas can describe the same local operator while using different base and fiber coordinates, and without an invariance statement their leading symbols might appear to disagree on overlaps. One must also exclude long-range kernel effects that are invisible in a near-diagonal oscillatory calculation. The formal result isolates the leading part that survives these coordinate changes and locality requirements.
[quotetheorem:7320]
[citeproof:7320]
The theorem is the point at which the local calculus becomes geometric, but its hypotheses are doing real work. Proper support cannot be removed: on $M=\mathbb R$, the kernel $K(x,y)=\rho(x)\rho(y-N)$, with $\rho\in C_c^\infty((-1,1))$ and $N$ arbitrarily large, is smoothing near the diagonal in the chart around $0$, but it sends input near $N$ into output near $0$; no local principal symbol near $x=y$ detects this long-range term. Compatible cutoffs are also necessary: if two overlapping charts are assigned the unrelated symbols $a(x,\xi)=|\xi|^m$ and $b(y,\eta)=2|\eta|^m$ for the same local operator region, the two formulas give different leading actions on the same oscillation unless the overlap comparison is imposed. The half-density normalization matters because under $y=\kappa(x)$, kernels on scalar functions acquire factors involving $|\det d\kappa|$; without half-densities or a fixed background density, those factors shift subprincipal data and prevent a coordinate-independent full symbol. Near-diagonal localization is equally essential: adding a kernel supported where $|x-y|>1$ changes the operator globally while leaving every oscillatory calculation near the diagonal unchanged. Thus the theorem identifies only the leading invariant, not a canonical full symbol.
It says that the leading term measures how an operator acts on high-frequency oscillations, and this measurement is intrinsic because the differential of the phase is intrinsic. The basic geometric differential operator should therefore have a principal symbol expressible without reference to a chart.
[example: Principal Symbol of the Semiclassical Laplacian]
Let $(M,g)$ be a Riemannian manifold and let $P_h=-h^2\Delta_g$ act locally on functions. In coordinates,
\begin{align*}
\Delta_g u=|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\partial_{x_j}u\bigr),
\end{align*}
with summation over repeated indices. Expanding the derivative by the product rule gives
\begin{align*}
-h^2\Delta_g u=-h^2g^{ij}\partial_{x_i}\partial_{x_j}u-h^2|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\bigr)\partial_{x_j}u.
\end{align*}
Using the semiclassical convention $hD_{x_i}=-ih\partial_{x_i}$, we have
\begin{align*}
(hD_{x_i})(hD_{x_j})u=(-ih)^2\partial_{x_i}\partial_{x_j}u=-h^2\partial_{x_i}\partial_{x_j}u.
\end{align*}
Therefore the second-order part of $P_h$ is
\begin{align*}
g^{ij}(x)(hD_{x_i})(hD_{x_j})u.
\end{align*}
By *Principal Symbol of a Semiclassical Differential Operator*, the order-two principal symbol is obtained from the degree-two part by replacing $hD_{x_i}$ with $\xi_i$, hence
\begin{align*}
p(x,\xi)=g^{ij}(x)\xi_i\xi_j.
\end{align*}
Equivalently,
\begin{align*}
p(x,\xi)=\sum_{i,j=1}^n g^{ij}(x)\xi_i\xi_j=|\xi|_{g^{-1}}^2.
\end{align*}
The remaining term contains only one derivative of $u$:
\begin{align*}
-h^2|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\bigr)\partial_{x_j}u=-ih\,|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\bigr)(hD_{x_j})u.
\end{align*}
It is therefore an $h$ times first-order semiclassical term, so it lies below the order-two principal symbol and does not change $p(x,\xi)=|\xi|_{g^{-1}}^2$.
[/example]
## Proper Support and Global Operators
The next problem is that a local oscillatory integral operator need not send compactly supported functions to functions whose support is controlled in the same chart. To patch operators globally, we impose a support condition that makes compositions with cutoffs and partitions of unity well-behaved.
[definition: Properly Supported Operator]
Let $M$ be a smooth manifold. An operator $A:C_c^\infty(M)\to C^\infty(M)$ is properly supported if both coordinate projections from $\operatorname{supp}K_A\subset M\times M$ to $M$ are proper maps, where $K_A$ is the Schwartz kernel of $A$.
[/definition]
Proper support ensures that $A$ extends from compactly supported test functions to smooth functions by localization: after multiplying the output by a compactly supported cutoff, only a compact part of the input is relevant. We now need a class of operators that combines this support condition with the requirement that every localized piece belongs to the Euclidean semiclassical calculus. The following definition packages those two requirements into the global class used on manifolds.
[definition: Semiclassical Pseudodifferential Operator on a Manifold]
Let $M$ be a smooth manifold. A properly supported operator $A_h:C_c^\infty(M)\to C^\infty(M)$ belongs to $\Psi_h^m(M)$ if for every pair of compactly supported coordinate cutoffs $\chi,\psi\in C_c^\infty(M)$ supported in coordinate neighbourhoods, the localized operator $\chi A_h\psi$ is, in the corresponding coordinates, a semiclassical pseudodifferential operator with symbol in $S^m$ up to an $O(h^\infty)$ smoothing remainder.
[/definition]
This definition is local, but it is designed so that all coordinate choices are tested by cutoffs. The remaining question is whether compatible local symbols can be assembled to produce such an operator, rather than merely recognized after the fact.
[quotetheorem:7321]
[citeproof:7321]
The theorem is the construction tool for the rest of the course, but it is not a uniqueness theorem for full symbols. Local finiteness is needed because otherwise infinitely many chart contributions could meet a fixed compact set: on $M=\mathbb R$, take intervals $U_j=(-1,1)$ repeated with the same chart and cutoffs all supported near $0$; the formal sum $\sum_j \chi_j\operatorname{Op}_h(1)\chi_j$ then contains infinitely many identical identity-type contributions near $0$ and does not define a finite local operator. The support condition is also essential. For instance, kernels of the form $K_j(x,y)=\rho(x)\rho(y-j)$ on $\mathbb R$ are smooth and compactly supported in each summand, but the locally finite sum over $j$ sends input from infinitely many distant regions into a fixed compact output region near $0$, so the projection of the total support to the second factor is not proper over compact output sets. Finally, overlap compatibility is exactly what prevents two local formulas from producing different leading symbols on the same covector; assigning $|\xi|^m$ on one chart and $3|\eta|^m$ on an overlapping chart cannot glue to a function on $T^*M$. What survives these choices canonically is the principal symbol; the full symbol and the particular quantization remain dependent on the partition, cutoffs, and coordinate conventions.
It lets us define operators by local formulas and then reason about their leading behaviour on the cotangent bundle. It also explains why the principal symbol is canonical while the full symbol depends on quantization choices. The circle gives a compact model where the patching is visible but the geometry remains one-dimensional.
[example: Quantization on the Circle Using Local Charts]
Let $M=S^1$, and choose two coordinate arcs $(U_j,\kappa_j)$ whose union is $S^1$. Write $t_j=\kappa_j(\theta)$ on $U_j$, and choose $\chi_j,\psi_j\in C_c^\infty(U_j)$ with $\chi_1+\chi_2=1$ and $\psi_j=1$ on a neighbourhood of $\operatorname{supp}\chi_j$. If $a\in S^m(T^*S^1)$, its local representative in the $j$th chart is
\begin{align*}
a_j(t_j,\tau_j)=a\bigl(\theta,\tau_j\,dt_j\bigr).
\end{align*}
The patched operator is
\begin{align*}
A_hu=\chi_1\operatorname{Op}_{h,\kappa_1}(a_1)(\psi_1u)+\chi_2\operatorname{Op}_{h,\kappa_2}(a_2)(\psi_2u).
\end{align*}
Since the cover is finite and each summand is localized inside one coordinate arc, the support condition in the *Globalization Theorem for Semiclassical Pseudodifferential Operators* is satisfied; hence $A_h\in\Psi_h^m(S^1)$.
On an overlap $U_1\cap U_2$, write $t_2=f(t_1)$ for the coordinate change. A covector can be written either as $\tau_1\,dt_1$ or as $\tau_2\,dt_2$, so
\begin{align*}
\tau_1\,dt_1=\tau_2\,dt_2.
\end{align*}
Since $dt_2=d(f(t_1))=f'(t_1)\,dt_1$, substitution gives
\begin{align*}
\tau_1\,dt_1=\tau_2 f'(t_1)\,dt_1.
\end{align*}
Equality of coefficients of $dt_1$ gives
\begin{align*}
\tau_1=f'(t_1)\tau_2.
\end{align*}
Therefore the two local symbols represent the same function on $T^*S^1$ because
\begin{align*}
a_2(f(t_1),\tau_2)=a\bigl(\theta,\tau_2\,dt_2\bigr)=a\bigl(\theta,f'(t_1)\tau_2\,dt_1\bigr)=a_1(t_1,f'(t_1)\tau_2).
\end{align*}
Thus the leading symbols glue to the original global symbol $a$. Changing the arcs or the cutoffs changes only the symbolic expansion terms involving derivatives of the cutoffs, which enter with an extra factor of $h$ and hence affect lower-order coefficients rather than the principal symbol.
[/example]
The use of cutoffs raises a natural concern: a partition of unity inserts many extra derivatives into local formulas. The next example isolates where those derivatives go in the symbol expansion.
[example: Partition Cutoffs Change Lower-Order Terms]
Let one patched summand be $\chi_j\operatorname{Op}_h(a_j)\psi_j$, where $a_j\in S^m$ and $\psi_j$ is multiplication by a smooth cutoff. The possible cutoff-dependent term is measured by moving $\psi_j$ past the local operator:
\begin{align*}
[\operatorname{Op}_h(a_j),\psi_j]=\operatorname{Op}_h(a_j)\psi_j-\psi_j\operatorname{Op}_h(a_j).
\end{align*}
For left semiclassical quantization, multiplication by $\psi_j(x)$ on the left has symbol $\psi_j(x)a_j(x,\xi)$. Multiplication by $\psi_j$ on the right is composition with the symbol $\psi_j(x)$, so the standard composition expansion gives
\begin{align*}
a_j\#\psi_j=a_j\psi_j+\frac{h}{i}\sum_{k=1}^n \partial_{\xi_k}a_j\,\partial_{x_k}\psi_j+h^2 r_j
\end{align*}
with $r_j\in S^{m-2}$, because each $\xi$-derivative lowers the symbol order by one. Hence the commutator has symbol
\begin{align*}
(a_j\#\psi_j)-\psi_j a_j=\frac{h}{i}\sum_{k=1}^n \partial_{\xi_k}a_j\,\partial_{x_k}\psi_j+h^2 r_j.
\end{align*}
Since $a_j\in S^m$, each $\partial_{\xi_k}a_j$ lies in $S^{m-1}$, while each $\partial_{x_k}\psi_j$ is smooth and compactly supported. Therefore
\begin{align*}
\frac{h}{i}\sum_{k=1}^n \partial_{\xi_k}a_j\,\partial_{x_k}\psi_j\in hS^{m-1}
\end{align*}
and
\begin{align*}
h^2r_j\in h^2S^{m-2}\subset hS^{m-1}.
\end{align*}
Thus $[\operatorname{Op}_h(a_j),\psi_j]\in h\Psi_h^{m-1}$.
Consequently, replacing partition cutoffs while keeping the same principal overlap data can only add terms built from derivatives of cutoffs, and every such derivative enters the symbolic expansion with at least one explicit factor of $h$ and one fewer power of $\xi$. The glued order-$m$ principal symbol is unchanged; only the subprincipal and lower coefficients depend on the particular partition.
[/example]
## Half-Densities, Functions, and Geometric Principal Symbols
The final issue in this chapter is that operators are often written on functions, but their kernels transform most naturally as densities. Half-densities provide a coordinate-free normalization in which adjoints, kernels, and principal symbols behave with fewer Jacobian corrections.
[definition: Half-Density]
Let $M$ be a smooth manifold. A half-density is a section of the complex line bundle $|\Omega_M|^{1/2}$ whose square transforms as a density under coordinate changes.
[/definition]
Half-densities let us integrate products of two half-densities without choosing an auxiliary volume form. If $u$ and $v$ are compactly supported half-densities, their product $u\overline{v}$ is a density and can be integrated over $M$. This makes half-densities the natural objects for kernels and formal adjoints, so we need the analogue of $\Psi_h^m(M)$ acting on this bundle rather than on scalar functions.
[definition: Pseudodifferential Operator on Half-Densities]
A semiclassical pseudodifferential operator of order $m$ on half-densities is a properly supported operator
\begin{align*}
A_h:C_c^\infty(M;|\Omega_M|^{1/2})\to C^\infty(M;|\Omega_M|^{1/2})
\end{align*}
whose localized coordinate representatives belong to the usual local class $\Psi_h^m$ after the standard coordinate trivialization of half-densities.
[/definition]
For functions, choosing a positive smooth density $\mu$ identifies a function $u$ with the half-density $u\mu^{1/2}$. This identification transfers the half-density calculus to functions, but lower-order terms now remember the density choice.
[remark: Functions Versus Half-Densities]
The principal symbol of an operator acting on functions is still a scalar function on $T^*M$ once a quantization convention is fixed. However, coordinate changes for kernels on functions include Jacobian factors that are absent in the half-density normalization. These factors do not change the order-$m$ symbol, but they influence subprincipal symbols and adjoint formulas.
[/remark]
This distinction is especially visible for geometric differential operators. Their leading symbols are intrinsic tensors, while their lower-order coordinate expressions depend on the chosen bundle, density, or connection. To connect the abstract pseudodifferential symbol map with familiar operators, we need the direct formula for the principal symbol of a semiclassical differential operator.
[quotetheorem:7322]
[citeproof:7322]
This theorem places differential operators inside the pseudodifferential symbol sequence and recovers the Laplacian example as a special case. The smooth bounded coefficient hypothesis ensures that differentiating the coefficients during symbolic manipulations remains within the same local symbol class. A concrete failure occurs already for $P_h=a(x)hD_x$ on $\mathbb R$ with $a(x)=|x|$: the formal leading expression $a(x)\xi$ is continuous but not a smooth symbol, and symbolic compositions would require derivatives of $a$ that are not smooth at $0$. The coordinate condition is also substantive. If $P_h=(hD_x)^2$ is written after the change $y=x^3$ on $(0,\infty)$, then $hD_x=3x^2 hD_y$, and the top part becomes $(3x^2)^2(hD_y)^2$ while additional first-order terms appear when derivatives hit $3x^2$. The degree-two coefficient has transformed as the symmetric tensor dual to covectors, giving the same quadratic form on $T^*M$; an arbitrary coordinatewise assignment of top coefficients would not satisfy this transformation law and would not define a global principal symbol. Lower-order terms, including derivatives of coefficients and density corrections, may change under reparametrisation, but they cannot alter the homogeneous degree-$m$ polynomial in the covector.
Once operators have a well-defined leading symbol, the next structural question is whether every possible principal symbol occurs and whether vanishing of that symbol is exactly the same as dropping to the next semiclassical order. This is a global statement, not only a chartwise statement: surjectivity requires patching local quantizations, and the kernel statement requires controlling all localized pieces at once. Compactness is the clean setting because a finite refinement removes the possibility that infinitely many lower-order remainders accumulate at infinity.
[quotetheorem:7323]
[citeproof:7323]
The exact sequence is the formal version of the principle used throughout microlocal analysis: the first invariant attached to an operator is its principal symbol, and after that symbol vanishes the operator drops by one semiclassical order with an extra factor of $h$. The equality of the kernel with $h\Psi_h^{m-1}(M)$ is stronger than saying that the leading coefficient is zero: the semiclassical convention records that the next possible contribution carries both one less symbolic order and one explicit power of $h$. Compactness is used to keep the patching finite after refinement, so no additional uniformity or proper-support condition at infinity has to be built into the statement. On a noncompact manifold, this can fail in a concrete way. Cover $\mathbb R$ by unit intervals and choose disjoint cutoffs $\chi_j$ near $j$; the operator
\begin{align*}
A_h=\sum_{j=1}^{\infty} j\,h\,\chi_j(x)hD_x\chi_j(x)
\end{align*}
has vanishing order-one principal symbol modulo $hS^0$ in each fixed chart, but the coefficients of the putative $h\Psi_h^0$ representative have symbol seminorms growing like $j$ and hence are not uniformly controlled on the noncompact manifold. Similarly, a sum of smoothing kernels $K_j(x,y)=\rho(x)\rho(y-j)$ is locally smoothing near every finite chart but is not properly supported, so it does not belong to the global residual class required for exactness. The noncompact symbol sequence therefore needs uniform symbol bounds and proper support built into the operator class.
This exact sequence is also the bridge to Hamiltonian dynamics and spectral theory: the principal symbol gives the Hamiltonian on $T^*M$ whose flow governs propagation of singularities, wave front sets for hyperbolic equations, and high-frequency eigenfunction behaviour. In later spectral asymptotics, the same principal symbol determines phase-space volumes such as those appearing in Weyl laws. A geometric Schrödinger operator shows how this bookkeeping separates the metric part from lower-order physics.
[example: Elliptic Geometric Operator]
Let $P_h=-h^2\Delta_g+V$ on a compact Riemannian manifold, with $V\in C^\infty(M)$ acting by multiplication. In local coordinates,
\begin{align*}
\Delta_g u=|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\partial_{x_j}u\bigr).
\end{align*}
Expanding the derivative gives
\begin{align*}
\Delta_g u=g^{ij}\partial_{x_i}\partial_{x_j}u+|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\bigr)\partial_{x_j}u.
\end{align*}
Since $hD_{x_i}=-ih\partial_{x_i}$, we have
\begin{align*}
(hD_{x_i})(hD_{x_j})u=(-ih)^2\partial_{x_i}\partial_{x_j}u=-h^2\partial_{x_i}\partial_{x_j}u.
\end{align*}
Therefore the second-order part of $-h^2\Delta_g$ is
\begin{align*}
g^{ij}(x)(hD_{x_i})(hD_{x_j})u.
\end{align*}
The remaining first-order part is
\begin{align*}
-h^2|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\bigr)\partial_{x_j}u=-ih\,|g|^{-1/2}\partial_{x_i}\bigl(|g|^{1/2}g^{ij}\bigr)(hD_{x_j})u.
\end{align*}
It has one factor $hD_{x_j}$ and an explicit factor of $h$, so it is below the order-two principal symbol. The potential term is
\begin{align*}
Vu=V(x)(hD_x)^0u,
\end{align*}
so it has differential order zero. By *Principal Symbol of a Semiclassical Differential Operator*, the order-two principal symbol is obtained from the degree-two part by replacing each $hD_{x_i}$ with $\xi_i$, hence
\begin{align*}
p(x,\xi)=g^{ij}(x)\xi_i\xi_j.
\end{align*}
Equivalently,
\begin{align*}
p(x,\xi)=\sum_{i,j=1}^n g^{ij}(x)\xi_i\xi_j=|\xi|_{g^{-1}}^2.
\end{align*}
Thus, when $P_h$ is regarded as an order-two operator, $V$ does not enter the principal symbol; it becomes visible only in a lower-order or rescaled effective symbol, so the chosen symbolic order determines which terms are leading.
[/example]
By the end of this chapter, the local semiclassical calculus has been converted into a global calculus on manifolds. The main invariant is the principal symbol on $T^*M$, while properly supported patching, partitions of unity, and half-density conventions provide the infrastructure needed to make local Fourier formulas coordinate-free.
With the manifold calculus established, the abstract theory becomes a practical tool for local microlocal estimates. The final chapter of this sequence uses ellipticity, parametrices, and localization to isolate the relevant phase-space region and extract concrete information about solutions.
# 10. Microlocal Elliptic Analysis as a Working Toolkit
This chapter turns the symbolic calculus from the previous chapters into a practical method for local estimates. The guiding question is: if an operator is elliptic only on part of phase space, how do we isolate that part, build a parametrix there, and conclude that a solution has no semiclassical singularities in that region? We will use microlocal partitions, nested cutoffs, and one capstone elliptic estimate for a semiclassical Schrödinger operator.
The point is not to introduce propagation of singularities. Elliptic analysis is the part of microlocal analysis that works without bicharacteristics: where the principal symbol stays away from zero, the equation itself gives control. This chapter packages that principle into a working toolkit that will be reused whenever the course needs to discard elliptic regions before studying the characteristic set.
## Microlocal Partitions of Unity and Localizing Estimates
Global estimates are rarely proved in one piece. The phase space $T^*\mathbb R^n$ contains low-frequency regions, elliptic regions, characteristic regions, and leftover transition regions, and each part asks for a different argument. The first task is to replace a single estimate by finitely many estimates after applying pseudodifferential cutoffs whose symbols add to $1$ on the relevant compact set.
[definition: Semiclassical Microlocal Partition]
Let $K \subset T^*\mathbb R^n$ be compact. A finite family of functions
\begin{align*}
\chi_j:T^*\mathbb R^n\to \mathbb C, \qquad \chi_j\in C_c^\infty(T^*\mathbb R^n), \qquad 1\le j\le N,
\end{align*}
is a semiclassical microlocal partition on $K$ if
\begin{align*}
\sum_{j=1}^N \chi_j(x,\xi) = 1
\end{align*}
for every $(x,\xi) \in K$.
[/definition]
The definition is deliberately finite because the course uses it on compact microlocal supports. After quantization, the identity is not exact on functions; instead it is exact microlocally, up to terms whose symbols vanish on the compact set being studied.
[example: Two Region Partition Near a Compact Set]
Let $K \subset T^*\mathbb R^n$ be compact and suppose $K\subset U_0\cup U_1$, where $U_0$ and $U_1$ are open. Choose $\rho_0,\rho_1\in C_c^\infty(T^*\mathbb R^n)$ with $\operatorname{supp}\rho_j\subset U_j$ near $K$ and with $\rho_0+\rho_1>0$ on some open neighbourhood $W$ of $K$. Choose $\eta\in C_c^\infty(W)$ with $\eta=1$ on a smaller neighbourhood of $K$, and define
\begin{align*}
\chi_j=\eta\frac{\rho_j}{\rho_0+\rho_1}
\end{align*}
on $\operatorname{supp}\eta$, extending by $0$ outside $W$. Then $\chi_j\in C_c^\infty(T^*\mathbb R^n)$ and $\operatorname{supp}\chi_j\subset U_j$ near $K$ because $\rho_j$ vanishes outside $U_j$ near the region where the quotient is used.
On the neighbourhood where $\eta=1$, the sum is
\begin{align*}
\chi_0+\chi_1=\eta\frac{\rho_0}{\rho_0+\rho_1}+\eta\frac{\rho_1}{\rho_0+\rho_1}
\end{align*}
and hence
\begin{align*}
\chi_0+\chi_1=\eta\frac{\rho_0+\rho_1}{\rho_0+\rho_1}=\eta=1.
\end{align*}
Thus $\chi_0+\chi_1=1$ on $K$. If $A_j=\operatorname{Op}_h(\chi_j)$, then $A_0u_h$ and $A_1u_h$ are the two microlocal pieces of $u_h$ associated to $U_0$ and $U_1$; near $K$ their symbols add to the identity symbol, so the original family is recovered microlocally by summing the two localized pieces, up to the usual negligible error from quantizing a microlocal identity.
[/example]
This example gives the construction, but estimates need a stability statement: after replacing $u_h$ by its microlocal pieces, the pieces must still recover the original family up to negligible error. The next lemma supplies exactly that replacement principle, using uniform boundedness of compactly supported order-zero cutoffs on semiclassical Sobolev spaces.
[quotetheorem:7324]
[proofunderconstruction:7324]
The lemma formalizes the phrase "work in one phase-space region at a time." It also explains why estimates may be proved after inserting cutoffs: the missing part is not uncontrolled; it is either another microlocal region or an $O(h^\infty)$ remainder. The compactness and finiteness hypotheses are essential here, since an infinite locally finite phase-space partition can introduce summability and uniform seminorm issues that are invisible in a single chart. The requirement that $\sum_j\chi_j=1$ on a neighbourhood of $K$, not merely pointwise on $K$, is also needed because quantization and composition move information by lower-order symbolic errors into a small surrounding region. If the equality held only on $K$, a family whose wavefront set approached the boundary of that equality region could leave an uncontrolled remainder after applying the quantized cutoffs. The lemma does not assert a global decomposition of $u_h$ on all of $T^*\mathbb R^n$; it is a statement about the chosen compact microlocal support, and that is exactly what lets the next section build parametrices with nested cutoffs.
[remark: Dependence on the Chosen Compact Set]
The constants in microlocal partition estimates depend on finitely many seminorms of the symbols and on the compact neighbourhoods used to separate supports. They do not depend on $h$ for $0<h\le h_0$. In applications, the compact set is fixed before estimates begin, so this dependence is harmless.
[/remark]
The next issue is that a single cutoff is not enough when commutators and parametrices enter. We need a larger cutoff that equals $1$ on the support of the smaller one, so that symbolic errors remain in a controlled annular region.
## Elliptic Cutoffs and Nested Supports
Suppose a symbol $p(x,\xi)$ is nonzero on the region where we want to control $u_h$. A natural idea is to divide by $p$ and quantize $1/p$, but this only makes sense on a set where $p$ stays away from zero. The cutoff construction records that division is performed locally and is protected by a slightly larger region.
[definition: Elliptic Region]
Let
\begin{align*}
p:T^*\mathbb R^n\times (0,h_0]\to \mathbb C, \qquad p\in S^m(T^*\mathbb R^n),
\end{align*}
be a semiclassical symbol and let $K \subset T^*\mathbb R^n$ be compact. The symbol $p$ is elliptic on $K$ if there exist $c>0$ and a neighbourhood $U$ of $K$ such that
\begin{align*}
|p(x,\xi;h)| \ge c\langle \xi\rangle^m
\end{align*}
for all $(x,\xi) \in U$ and all $0<h\le h_0$.
[/definition]
Ellipticity is a lower bound, not a global invertibility statement. To use that lower bound inside a composition, the estimate must distinguish the smaller region where the conclusion is measured from the larger region where division by $p$ is allowed; this motivates the following support notation.
[definition: Nested Cutoffs]
For functions
\begin{align*}
\chi_0,\chi_1:T^*\mathbb R^n\to \mathbb C, \qquad \chi_0,\chi_1 \in C_c^\infty(T^*\mathbb R^n),
\end{align*}
write $\chi_0 \prec \chi_1$ if $\chi_1=1$ on a neighbourhood of $\operatorname{supp}\chi_0$.
[/definition]
The notation $\chi_0\prec \chi_1$ is a microlocal version of saying that $\chi_1$ is a buffer around $\chi_0$. The smaller cutoff is where the final estimate is measured; the larger cutoff is where the equation is allowed to be used.
[example: Building Cutoffs Near a Compact Phase-Space Set]
Let $K\subset U\subset T^*\mathbb R^n$, with $K$ compact and $U$ open. Choose open sets $V_0,V_1$ such that $K\subset V_0$, $\overline{V_0}\subset V_1$, and $\overline{V_1}\subset U$. By the smooth cutoff lemma on $\mathbb R^{2n}$, choose $\chi_0\in C_c^\infty(V_1)$ with $\chi_0=1$ on a neighbourhood of $K$ and choose $\chi_1\in C_c^\infty(U)$ with $\chi_1=1$ on a neighbourhood of $\overline{V_1}$.
Since $\operatorname{supp}\chi_0\subset \overline{V_1}$ and $\chi_1=1$ on a neighbourhood of $\overline{V_1}$, it follows that $\chi_1=1$ on a neighbourhood of $\operatorname{supp}\chi_0$. Hence $\chi_0\prec\chi_1$. Also,
\begin{align*}
\operatorname{supp}\chi_0\subset \overline{V_1}\subset U
\end{align*}
and
\begin{align*}
\operatorname{supp}\chi_1\subset U.
\end{align*}
Thus both cutoffs are supported inside the elliptic neighbourhood $U$, while $\chi_1$ is identically $1$ around the smaller support where $\chi_0$ measures the final estimate. The open region between $\operatorname{supp}\chi_0$ and $T^*\mathbb R^n\setminus\{\chi_1=1\}$ is the buffer in which symbolic commutator and parametrix errors can occur without reaching the region selected by $\chi_0$.
[/example]
Nested supports are especially useful when composing a parametrix with an operator. The result below is the local estimate that turns ellipticity into regularity, with the larger cutoff carrying the data term.
[quotetheorem:7325]
[citeproof:7325]
The estimate should be read as a gain of $m$ semiclassical derivatives in the elliptic region. If $P_hu_h$ is very small there, then $u_h$ is very small after applying the smaller cutoff. Ellipticity must hold on the larger cutoff, not merely on $\operatorname{supp}\chi_0$, because the parametrix is composed with $P_h$ before the final localization and symbolic products sample a neighbourhood of the support. If $p$ vanishes inside $\operatorname{supp}\chi_1\setminus\operatorname{supp}\chi_0$, the term $A_1P_hu_h$ may fail to control the errors produced while moving cutoffs through the parametrix. The nesting hypothesis is equally necessary: if $\chi_1$ is not $1$ near $\operatorname{supp}\chi_0$, a commutator can create a contribution exactly where the estimate is trying to prove control. The negative Sobolev remainder is not a defect in ellipticity; it records that a finite symbolic parametrix leaves a smoothing operator, and polynomial or a priori Sobolev bounds are needed later to convert that smoothing gain into $O(h^\infty)$ control.
[example: Elliptic Equation Removes Wavefront]
Assume $P_h=\operatorname{Op}_h(p)$ is elliptic on an open set $U\subset T^*\mathbb R^n$ and $P_hu_h=O_{C^\infty}(h^\infty)$ microlocally in $U$. Fix $K\Subset U$, and choose $\chi_0,\chi_1\in C_c^\infty(T^*\mathbb R^n)$ with $\chi_0\prec\chi_1$, $\chi_0=1$ on a neighbourhood of $K$, and $\operatorname{supp}\chi_1\subset U$. Put $A_j=\operatorname{Op}_h(\chi_j)$.
For any Sobolev order $s$ and any target power $M$, apply the *Nested Cutoff Estimate* with a larger remainder order $L$ to get constants $C_{s,L}$ and $N_{s,L}$ such that
\begin{align*}
\|A_0u_h\|_{H_h^{s+m}}\le C_{s,L}\|A_1P_hu_h\|_{H_h^s}+C_{s,L}h^L\|u_h\|_{H_h^{-N_{s,L}}}.
\end{align*}
Since $\operatorname{supp}\chi_1\subset U$ and $P_hu_h=O_{C^\infty}(h^\infty)$ microlocally in $U$, the first term satisfies
\begin{align*}
\|A_1P_hu_h\|_{H_h^s}=O(h^M)
\end{align*}
for every $M$. In the wavefront-set setting the family $u_h$ is polynomially bounded in a sufficiently negative Sobolev norm, so for some $B$,
\begin{align*}
\|u_h\|_{H_h^{-N_{s,L}}}\le C h^{-B}.
\end{align*}
Choose $L\ge M+B$. Then
\begin{align*}
h^L\|u_h\|_{H_h^{-N_{s,L}}}\le C h^{L-B}\le C h^M.
\end{align*}
Thus $\|A_0u_h\|_{H_h^{s+m}}=O(h^M)$ for every $s$ and every $M$, which is exactly
\begin{align*}
\operatorname{Op}_h(\chi_0)u_h=O_{C^\infty}(h^\infty).
\end{align*}
Because $\chi_0=1$ near $K$, no point of $K$ lies in $\operatorname{WF}_h(u_h)$.
[/example]
This example is the standard elliptic regularity argument in semiclassical language. It says that microlocal singularities can only remain where the principal symbol may vanish, provided the equation is negligible away from that set.
## The Schrödinger Operator in an Elliptic Region
We now specialize the general toolkit to the model operator that motivates much of semiclassical analysis:
\begin{align*}
P_h=-h^2\Delta+V(x)-E.
\end{align*}
The question is what an approximate solution $P_hu_h=O(h^\infty)$ can do away from the energy surface. Elliptic analysis answers: nothing microlocally visible can remain where $|\xi|^2+V(x)-E$ is bounded away from zero.
[definition: Classical Schrödinger Symbol]
Let $V\in C^\infty(\mathbb R^n;\mathbb R)$ and $E\in\mathbb R$. The classical Schrödinger symbol associated to the operator
\begin{align*}
P_h:H_h^2(\mathbb R^n)\to L^2(\mathbb R^n), \qquad P_hu=-h^2\Delta u+(V-E)u,
\end{align*}
is the function
\begin{align*}
p:T^*\mathbb R^n\to\mathbb R, \qquad (x,\xi)\mapsto |\xi|^2+V(x)-E.
\end{align*}
The characteristic set at energy $E$ is
\begin{align*}
\Sigma_E=\{(x,\xi)\in T^*\mathbb R^n: p(x,\xi)=0\}.
\end{align*}
[/definition]
The characteristic set is the only phase-space region where the principal symbol fails to provide a local inverse. The theorem below is the capstone estimate: approximate eigenfunctions have semiclassical wavefront set contained in $\Sigma_E$, before any propagation result is invoked.
[quotetheorem:7326]
[proofunderconstruction:7326]
The theorem separates elliptic localization from propagation. It does not say how singularities move on $\Sigma_E$; it says that outside $\Sigma_E$ there are no singularities left to move. The quantitative lower bound is necessary because the parametrix contains division by $p$; if $p$ approaches $0$ along the tested region, the inverse symbol can lose uniform bounds as $h\to0$. Compactness is the mechanism that turns pointwise nonvanishing of $p$ away from $\Sigma_E$ into a uniform constant $c>0$ and lets the cutoffs have controlled seminorms. Polynomial boundedness of $u_h$ is also part of the conclusion: a smoothing remainder applied to an exponentially large family, for instance $\|u_h\|_{L^2}\sim e^{1/h}$, need not be $O(h^\infty)$ even when the residual operator carries high powers of $h$. Thus the theorem is an exclusion result for moderately bounded quasimodes on compact elliptic regions, and the next examples show how that exclusion becomes the first reduction before propagation on the energy surface.
[example: Removing Wavefront Outside the Energy Surface]
Let $p(x,\xi)=|\xi|^2+V(x)-E$ and let $d=\operatorname{dist}(K,\Sigma_E)>0$. Choose $0<\delta<d/2$ and set
\begin{align*}
K_\delta=\{(x,\xi):\operatorname{dist}((x,\xi),K)\le \delta\}.
\end{align*}
Then $K_\delta$ is compact and $K_\delta\cap\Sigma_E=\varnothing$, because any point of $K_\delta\cap\Sigma_E$ would have distance at most $\delta<d$ from $K$ while lying in $\Sigma_E$. Hence $p$ has no zero on $K_\delta$. Since $p$ is continuous and $K_\delta$ is compact, the minimum
\begin{align*}
c_K=\min_{(x,\xi)\in K_\delta}|p(x,\xi)|
\end{align*}
exists and satisfies $c_K>0$. Therefore
\begin{align*}
\big||\xi|^2+V(x)-E\big|=|p(x,\xi)|\ge c_K
\end{align*}
on the neighbourhood $\{(x,\xi):\operatorname{dist}((x,\xi),K)<\delta\}$ of $K$.
Assume $\|u_h\|_{L^2}\le Ch^{-M_0}$ and $P_hu_h=O_{C^\infty}(h^\infty)$ microlocally on the compact spatial region containing the projection of this neighbourhood. The hypotheses of the *Elliptic Localization Theorem for Semiclassical Schrödinger Operators* are then satisfied on $K$, so
\begin{align*}
K\cap\operatorname{WF}_h(u_h)=\varnothing.
\end{align*}
Thus, in that compact region, any point of $\operatorname{WF}_h(u_h)$ must lie where $p(x,\xi)=0$, equivalently on the energy surface
\begin{align*}
|\xi|^2+V(x)=E.
\end{align*}
[/example]
This is often the first microlocal step in studying eigenfunctions. Before asking how mass travels along Hamiltonian trajectories, one restricts attention to the shell where the Hamiltonian vanishes.
[example: Separating Low-Frequency, Elliptic, and Residual Regions]
Fix a compact spatial set $X\subset\mathbb R^n$ and $R>0$, and write
\begin{align*}
K_R=\{(x,\xi):x\in X,\ |\xi|\le R\}.
\end{align*}
Choose $\varepsilon>0$ and a smooth function $\theta:[0,\infty)\to[0,1]$ such that $\theta(t)=1$ for $0\le t\le\varepsilon$ and $\theta(t)=0$ for $t\ge 2\varepsilon$. On a neighbourhood of $K_R$, define
\begin{align*}
\chi_{\mathrm{char}}(x,\xi)=\theta(|p(x,\xi)|).
\end{align*}
Choose $\chi_{\mathrm{ell}}\in C_c^\infty(T^*\mathbb R^n)$ supported where $|p|\ge \varepsilon$ and equal to $1$ on the part of $K_R$ where $|p|\ge 2\varepsilon$, after inserting a harmless spatial-frequency cutoff equal to $1$ near $K_R$. Finally define $\chi_{\mathrm{res}}$ on $K_R$ by
\begin{align*}
\chi_{\mathrm{res}}=1-\chi_{\mathrm{char}}-\chi_{\mathrm{ell}}.
\end{align*}
Then on $K_R$,
\begin{align*}
\chi_{\mathrm{char}}+\chi_{\mathrm{ell}}+\chi_{\mathrm{res}}=1.
\end{align*}
The supports have the intended meanings. Since $\chi_{\mathrm{char}}=1$ when $|p|\le\varepsilon$, it captures a fixed neighbourhood of the characteristic set $\Sigma_E\cap K_R$. Since $\operatorname{supp}\chi_{\mathrm{ell}}\subset\{|p|\ge\varepsilon\}$, the Schrödinger symbol is elliptic on the elliptic piece with lower bound
\begin{align*}
|p(x,\xi)|\ge\varepsilon.
\end{align*}
Thus, if $P_hu_h=O_{C^\infty}(h^\infty)$ microlocally there and $u_h$ is polynomially bounded, the *Elliptic Localization Theorem for Semiclassical Schrödinger Operators* gives
\begin{align*}
\operatorname{Op}_h(\chi_{\mathrm{ell}})u_h=O_{C^\infty}(h^\infty).
\end{align*}
The residual symbol is supported only where neither cutoff has already settled to its constant value, hence inside the transition shell
\begin{align*}
\operatorname{supp}\chi_{\mathrm{res}}\cap K_R\subset\{(x,\xi)\in K_R:\varepsilon\le |p(x,\xi)|\le 2\varepsilon\}.
\end{align*}
Choosing $\varepsilon$ smaller confines this shell to a narrower fixed phase-space neighbourhood of $\Sigma_E\cap K_R$. Therefore the elliptic part is removed by the equation, while the characteristic and transition pieces are the only pieces left for later propagation estimates.
[/example]
The separation in this example is not a propagation argument. It is a bookkeeping device that ensures every later estimate knows exactly which phase-space region it is addressing.
## Practical Checklist for Elliptic Microlocal Estimates
When applying elliptic microlocal analysis, the main risk is using a parametrix outside its domain of ellipticity. The following checklist records the order of operations used throughout the course.
[explanation: Elliptic Estimate Workflow]
Start by naming the compact phase-space set $K$ where the conclusion is required. Verify a quantitative lower bound for the relevant principal symbol on an open neighbourhood of $K$. Then choose nested cutoffs $\chi_0\prec\chi_1$ with $\chi_0=1$ near $K$ and $\operatorname{supp}\chi_1$ still inside the elliptic neighbourhood.
After the cutoffs are fixed, construct the parametrix only on the support of the larger cutoff. Apply the nested cutoff estimate to convert control of $P_hu_h$ near $\operatorname{supp}\chi_1$ into control of $u_h$ near $\operatorname{supp}\chi_0$. Finally, use polynomial boundedness or an a priori Sobolev bound to make the smoothing remainder smaller than the required power of $h$.
[/explanation]
This workflow is the reason elliptic regularity is robust in semiclassical problems. The constants and remainders depend on the chosen cutoffs, but once those cutoffs are fixed, the argument is uniform as $h\to0$.
[remark: What Elliptic Analysis Does Not Prove]
Elliptic estimates do not describe transport along the characteristic set, do not identify incoming or outgoing pieces, and do not replace energy estimates. Their role is to remove the phase-space regions where the operator has an inverse. The remaining characteristic region is the arena for propagation, defect measures, and finer dynamical arguments.
[/remark]
## Connections and Further Reading
These notes are the microlocal foundation for propagation of singularities, semiclassical defect measures, scattering theory, and spectral asymptotics. The common thread is that the calculus replaces a differential or integral equation by geometry on $T^*M$: ellipticity removes regions where the principal symbol is invertible, while characteristic regions retain the Hamiltonian dynamics that later estimates must follow.
The next natural topics are propagation estimates along bicharacteristics, positive commutator methods, Fourier integral operators, and Egorov-type theorems. For Schrödinger operators, the elliptic localization result here is only the first step: it confines semiclassical wavefront set to the energy surface, but transport of mass along the Hamilton flow requires a propagation theorem. For boundary value problems or noncompact manifolds, one must also add boundary calculi, escape functions, or weighted spaces to keep the same symbolic ideas under control.
At the level of analysis, semiclassical Sobolev spaces connect these notes to standard elliptic regularity, while the wave packet examples connect them to coherent states and phase-space distributions. The sharp Garding and Fefferman--Phong estimates are the bridge from symbolic positivity to quantitative energy estimates. They explain why microlocal sign conditions produce coercive inequalities only after accepting lower-order semiclassical losses.
Contents
- Introduction
- Why a Semiclassical Parameter Appears
- Position, Frequency, and Phase Space
- Uniform Estimates as the Main Language
- What the First Part of the Course Builds
- 1. Semiclassical Scaling and Model Oscillation
- Uniform Estimates for Parameter-Dependent Families
- The Semiclassical Fourier Transform
- Semiclassical Derivatives and Frequency Scaling
- Uniform Schwartz Control
- Nonstationary Semiclassical Phase
- Wave Packets and Phase-Space Concentration
- 2. Semiclassical Symbol Classes
- Measuring Growth in Phase Space
- Mildly Singular Semiclassical Classes
- Classical Symbols And Leading Terms
- Asymptotic Expansions And Summation
- Compact Support And Microlocal Localization
- 3. Quantization on Euclidean Space
- Kohn--Nirenberg Quantization and Operator Kernels
- Weyl Quantization and Symmetric Ordering
- General Tau-Quantization and Conversion Between Conventions
- 4. The Semiclassical Pseudodifferential Calculus
- Composition and the Symbolic Product
- Weyl Quantization and the Moyal Product
- Commutators, Adjoints, and the Algebra $\Psi_h^m$
- 5. Mapping Properties and Semiclassical Sobolev Spaces
- Semiclassical Sobolev Spaces
- Uniform $L^2$ Boundedness
- Order Reduction and Sobolev Mapping
- Proper Support and Localization
- Pseudolocality
- 6. Ellipticity and Parametrices
- Elliptic Symbols and Characteristic Sets
- Parametrices by Symbolic Inversion
- Microlocal Elliptic Estimates
- Elliptic Regularity for Semiclassical Pseudodifferential Operators
- 7. Semiclassical Wavefront Sets
- Testing Phase Space by Annihilators
- Fourier Tests in Euclidean Charts
- Fiber Infinity and Microsupport
- 8. Positivity and Sharp Garding Estimates
- The Positivity Problem for Quantization
- Weyl Quantization and Symbolic Square Roots
- Sharp Garding Inequality
- Fefferman-Phong Type Lower Bounds
- Role in Microlocal Energy Estimates
- 9. Pseudodifferential Operators on Manifolds
- Coordinate Changes and Principal Symbols on the Cotangent Bundle
- Proper Support and Global Operators
- Half-Densities, Functions, and Geometric Principal Symbols
- 10. Microlocal Elliptic Analysis as a Working Toolkit
- Microlocal Partitions of Unity and Localizing Estimates
- Elliptic Cutoffs and Nested Supports
- The Schrödinger Operator in an Elliptic Region
- Practical Checklist for Elliptic Microlocal Estimates
- Connections and Further Reading
Semiclassical Analysis I: Symbols, Quantization, and Microlocal Foundations
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Created by admin on 6/17/2026 | Last updated on 6/17/2026
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