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]