Duistermaat-Guillemin Wave Trace Formula for Clean Closed Geodesics

Theorem

Edit Issues Pull Requests Attributions Admin

Let $(M,g)$ be a compact smooth Riemannian manifold without boundary, and let $\Delta_g \geq 0$ denote the nonnegative Laplace-Beltrami operator on complex-valued functions. Let \begin{align*} P := \sqrt{\Delta_g} \end{align*} be defined by the spectral theorem. If \begin{align*} 0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 \leq \cdots \end{align*} is the spectrum of $\Delta_g$, repeated with multiplicity, define the half-wave trace distribution \begin{align*} \Theta \in \mathcal{D}'(\mathbb{R}) \end{align*} by \begin{align*} \Theta(t) := \operatorname{Tr}(e^{-itP}) = \sum_{j=0}^{\infty} e^{-it\sqrt{\lambda_j}} \end{align*} in the sense of distributions. Let \begin{align*} G^t:S^*M \to S^*M \end{align*} denote the geodesic flow on the unit cosphere bundle. If $t_0 \in \mathbb{R}\setminus\{0\}$ and $\Theta$ is not smooth in any open neighbourhood of $t_0$, then there exists $\rho \in S^*M$ such that \begin{align*} G^{t_0}(\rho)=\rho. \end{align*} Equivalently, $|t_0|$ is the length of a closed geodesic on $M$. Conversely, fix $t_0 \in \mathbb{R}\setminus\{0\}$ and define \begin{align*} F_{t_0}:=\{\rho \in S^*M : G^{t_0}(\rho)=\rho\}. \end{align*} Assume that $F_{t_0}$ is clean, meaning that $F_{t_0}$ is a smooth submanifold of $S^*M$ and, for every $\rho \in F_{t_0}$, \begin{align*} T_\rho F_{t_0}=\ker(dG^{t_0}_\rho-I). \end{align*} Then each connected component $C$ of $F_{t_0}$ determines, after microlocal localization near $C$, a Duistermaat-Guillemin local contribution to $\Theta$ near $t=t_0$. This contribution is a classical conormal oscillatory distribution at $t_0$, with principal coefficient equal to the standard clean wave-trace density of Duistermaat and Guillemin, including the Maslov factor and the linearized Poincare-map data. If this principal clean wave-trace coefficient is nonzero for $C$, then the localized contribution from $C$ is genuinely singular at $t=t_0$. The singular contributions from different connected components may cancel in the unlocalized total trace.

Discussion

No discussion available for this theorem.

Proof

[proofplan] The proof is an application of the Duistermaat-Guillemin trace formula for the half-wave group. Microlocally, $e^{-it\sqrt{\Delta_g}}$ is a Fourier integral operator whose canonical relation is the graph of the homogeneous geodesic flow; taking the trace intersects this canonical relation with the diagonal. This intersection condition is precisely the fixed-point condition for $G^{t_0}$ on $S^*M$, which gives the singular support inclusion. Under the clean fixed-point hypothesis, clean stationary phase along each connected component of the fixed set gives the stated conormal oscillatory contribution and its principal clean wave-trace coefficient. [/proofplan] [step:Invoke the Duistermaat-Guillemin trace theorem for the half-wave group] We use the Duistermaat-Guillemin trace formula for the half-wave propagator, in the standard half-density and Maslov conventions of Duistermaat and Guillemin. This is an external result not yet in the wiki: Duistermaat-Guillemin trace formula for the half-wave group. Let \begin{align*} U(t):=e^{-itP}:C^\infty(M) \to C^\infty(M) \end{align*} denote the half-wave propagator. The microlocal construction of $U(t)$ states that, away from the zero section in $T^*M$, the Schwartz kernel of $U(t)$ is a Fourier integral distribution associated with the canonical relation \begin{align*} \Gamma_t := \{(x,\xi;y,\eta)\in T^*M\setminus 0 \times T^*M\setminus 0 : (x,\xi)=\Phi^t(y,\eta)\}, \end{align*} where \begin{align*} \Phi^t:T^*M\setminus 0 \to T^*M\setminus 0 \end{align*} is the homogeneous Hamiltonian flow of the principal symbol $|\xi|_g$ of $P$. Taking the trace means restricting the Schwartz kernel to the diagonal \begin{align*} \Delta_M:=\{(x,x):x\in M\}\subset M\times M \end{align*} and integrating against the Riemannian volume measure $d\operatorname{vol}_g(x)$. The Fourier integral [trace theorem](/theorems/60) says that the trace can be singular at a nonzero time $t_0$ only if the canonical relation $\Gamma_{t_0}$ meets the conormal relation of the diagonal. Equivalently, there must exist $(x,\xi)\in T^*M\setminus 0$ such that \begin{align*} \Phi^{t_0}(x,\xi)=(x,\xi). \end{align*} [guided] The operator \begin{align*} U(t):=e^{-itP}:C^\infty(M) \to C^\infty(M) \end{align*} is the object whose trace defines $\Theta$. The microlocal input is the following external theorem, not yet in the wiki: the half-wave propagator is a Fourier integral operator associated with the graph of the homogeneous geodesic flow. In symbols, away from the zero section of $T^*M$, the canonical relation of the Schwartz kernel of $U(t)$ is \begin{align*} \Gamma_t := \{(x,\xi;y,\eta)\in T^*M\setminus 0 \times T^*M\setminus 0 : (x,\xi)=\Phi^t(y,\eta)\}, \end{align*} where \begin{align*} \Phi^t:T^*M\setminus 0 \to T^*M\setminus 0 \end{align*} is the Hamiltonian flow of the principal symbol $|\xi|_g$. Why does tracing force a fixed point? The Schwartz kernel of $U(t)$ is a distribution on $M\times M$. The trace restricts that kernel to the diagonal \begin{align*} \Delta_M:=\{(x,x):x\in M\} \end{align*} and then integrates over $M$ with respect to $d\operatorname{vol}_g(x)$. The trace theorem for Fourier integral operators says that a singularity can survive this diagonal restriction only when the canonical relation of the kernel meets the diagonal canonical relation. For $\Gamma_{t_0}$, this intersection condition says that for some nonzero covector $(x,\xi)\in T^*M\setminus 0$, the point obtained by flowing for time $t_0$ is again the same covector: \begin{align*} \Phi^{t_0}(x,\xi)=(x,\xi). \end{align*} Thus the trace theorem gives the key implication: if no nonzero covector is fixed by the homogeneous geodesic flow at time $t_0$, then the diagonal pullback and integration produce a smooth distribution near $t_0$. Taking the contrapositive, if $\Theta$ is not smooth in any neighbourhood of $t_0$, then such a fixed nonzero covector must exist. [/guided] [/step] [step:Pass from homogeneous fixed covectors to fixed points on the unit cosphere bundle] The Hamiltonian $|\xi|_g$ is positively homogeneous of degree $1$ in $\xi$, and its flow preserves the level set \begin{align*} S^*M:=\{(x,\xi)\in T^*M:|\xi|_g=1\}. \end{align*} If $(x,\xi)\in T^*M\setminus 0$ satisfies $\Phi^{t_0}(x,\xi)=(x,\xi)$, define \begin{align*} \rho := (x,|\xi|_g^{-1}\xi)\in S^*M. \end{align*} By homogeneity of the Hamiltonian flow, \begin{align*} G^{t_0}(\rho)=\rho. \end{align*} Therefore non-smoothness of $\Theta$ at $t_0\neq 0$ implies the existence of a fixed point of $G^{t_0}$ on $S^*M$. [/step] [step:Identify fixed cosphere points with closed geodesics of length $|t_0|$] Let $\rho=(x,\xi)\in S^*M$ satisfy $G^{t_0}(\rho)=\rho$. Let \begin{align*} \gamma_\rho:\mathbb{R}\to M \end{align*} be the unit-speed geodesic determined by the initial covector $\rho$ through the metric identification $T_x^*M\cong T_xM$. The equation $G^{t_0}(\rho)=\rho$ says that both the base point and the unit covector return after time $t_0$. Hence $\gamma_\rho$ is periodic with period $|t_0|$, and its length over one period is $|t_0|$ because it has unit speed. Conversely, if $\gamma:\mathbb{R}\to M$ is a unit-speed closed geodesic of period $|t_0|$, then the metric dual covector to $\dot{\gamma}(0)$ determines a point $\rho\in S^*M$ fixed by $G^{t_0}$, after reversing the orientation of $\gamma$ if $t_0<0$. Thus the fixed-point condition is equivalent to $|t_0|$ being the length of a closed geodesic. [/step] [step:Localize microlocally near a clean connected component of the fixed set] Assume now that $F_{t_0}$ is clean, and let $C$ be a connected component of $F_{t_0}$. Choose a properly supported pseudodifferential microlocal cutoff \begin{align*} A_C:C^\infty(M)\to C^\infty(M) \end{align*} whose principal symbol is equal to $1$ in a conic neighbourhood of $C$ in $T^*M\setminus 0$ and whose wavefront support is contained in a sufficiently small conic neighbourhood of $C$ that meets no other component of $F_{t_0}$. The localized trace distribution \begin{align*} \Theta_C(t):=\operatorname{Tr}(A_C e^{-itP}) \end{align*} is well-defined microlocally near $t=t_0$, modulo a distribution smooth near $t_0$. The clean hypothesis says precisely that the phase critical set for the localized trace is the smooth manifold $C$ and that the normal Hessian is nondegenerate transverse to $C$. Therefore the clean [stationary phase theorem](/theorems/8198) for Fourier integral traces applies to $\Theta_C$. [guided] The phrase “the contribution of $C$” is microlocal. It is not a canonical global splitting of the full trace unless we first choose a cutoff. Let $C$ be a connected component of \begin{align*} F_{t_0}:=\{\rho\in S^*M:G^{t_0}(\rho)=\rho\}. \end{align*} Choose a properly supported pseudodifferential operator \begin{align*} A_C:C^\infty(M)\to C^\infty(M) \end{align*} whose principal symbol equals $1$ near $C$ and whose wavefront support is contained in a small conic neighbourhood of $C$ avoiding the other connected components of $F_{t_0}$. Then \begin{align*} \Theta_C(t):=\operatorname{Tr}(A_C e^{-itP}) \end{align*} isolates the part of the trace coming from bicharacteristics near $C$, modulo terms smooth near $t_0$. Why is the clean hypothesis exactly the right condition? The trace is computed by restricting the Fourier integral kernel of $e^{-itP}$ to the diagonal and then applying stationary phase in the phase variables. The critical points of that phase are exactly the fixed points of the geodesic flow. Near the component $C$, the critical set is therefore $C$ itself. The stated cleanliness condition, \begin{align*} T_\rho F_{t_0}=\ker(dG^{t_0}_\rho-I), \end{align*} says that the tangent directions to the critical set are precisely the degenerate directions of the linearized fixed-point equation. Equivalently, there is no additional degeneracy transverse to $C$. This is the hypothesis required by clean stationary phase: the Hessian is nondegenerate in the normal directions to the critical manifold. Therefore the clean stationary phase theorem applies to the localized trace $\Theta_C$. It produces an asymptotic expansion determined by the phase, the principal symbol of the half-wave propagator, the Maslov bundle, and the determinant data of the linearized return map transverse to $C$. [/guided] [/step] [step:Apply clean stationary phase to obtain the conormal wave-trace contribution] By clean stationary phase applied to the localized Fourier integral trace, $\Theta_C$ is, modulo a distribution smooth near $t_0$, a classical conormal oscillatory distribution associated with the point $\{t_0\}\subset\mathbb{R}$. Its principal coefficient is the Duistermaat-Guillemin clean wave-trace density on $C$, computed from the principal half-wave symbol, the Maslov factor, and the transverse linearized Poincare-map data. More explicitly, the transverse factor is determined by the [linear map](/page/Linear%20Map) induced by \begin{align*} dG^{t_0}_\rho-I:T_\rho(S^*M)\to T_\rho(S^*M) \end{align*} on the normal space \begin{align*} N_\rho C:=T_\rho(S^*M)/T_\rho C, \end{align*} while the phase contribution is recorded by the Maslov factor in the Duistermaat-Guillemin convention. These data combine to give the standard principal clean density along $C$. [/step] [step:Deduce genuine localized singularity from a nonzero principal coefficient] If the principal clean wave-trace coefficient of $C$ is nonzero, then the leading homogeneous term in the classical conormal expansion of $\Theta_C$ is nonzero. A classical conormal distribution with nonzero leading term at $\{t_0\}$ cannot be smooth in any neighbourhood of $t_0$, because smoothness would force all singular homogeneous terms in its conormal expansion to vanish. Hence the microlocal contribution from $C$ is genuinely singular at $t=t_0$. The full trace $\Theta$ is obtained, microlocally near $t_0$, by summing the localized contributions over the connected components of $F_{t_0}$, up to a smooth remainder. Since distributions with the same conormal order and phase may have opposite principal coefficients, different connected components can cancel in the total trace. This proves both the singular-support implication and the clean closed-geodesic contribution statement. [/step]

What brings you to Androma?

Start with a route through the knowledge graph.