What does the flow of the principal symbol of the differential operator tell us about the PDE?

Question

Disclaimer: Let me apologize in advance for asking this slightly vague question

Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there's the obvious linear PDE: $$P(f)=0$$

Naturally associated with $P$ we also have a hamiltonian system $$\Phi= (T^*M, \omega,H=\sigma(P))$$ where the symplectic form $\omega$ is the standard one and principal symbol $\sigma(P)$ of $P$ is taken as the hamiltonian.

Question: What does the dynamics of $\Phi$ tell us about the original differential equation?

Obviously since we are only considering the principal symbol we won't get a terrible amount of information. On the other hand we are not taking only the zero locus $\{ \sigma(P)=0\}$ (AKA the characteristic variety / set ) so one might hope that we can at least find some information in $\Phi$ which is not already present in the geometry of $\{\sigma(P)=0\}$.

Answer 1

A bit more advanced thing: Consider a first order classical pseudodifferential operator $P$ on a compact manifold $M$, which is elliptic and self-adjoint and its principal symbol is positive. Think again $\sqrt{\Delta}$ on a compact Riemannian manifold $(M,g)$.

By Sobolev embedding it has compact resolvent, so the spectrum is purely discrete and by positivity the spectrum is bounded from below. Set $N(\lambda) = \#\{ \lambda_j \colon \lambda_j < \lambda\}$ the counting function. It is well-known (cf. Hörmander) that the counting function has asymptotic behaviour $$N(\lambda) = \gamma_1\lambda^d + O(\lambda^{d-1})$$ for a constant $\gamma_1$ depending on the principal symbol.

Duistermaat and Guillemin (1975) showed if the set of periodic orbits of the Hamiltonian flow (restricted to a energy surface) has measure zero then the Weyl asymptotic has a second term: $$N(\lambda) = \gamma_1 \lambda^d + \gamma_2 \lambda^{d-1} + o(\lambda^{d-1}).$$

This argument is based (more or less) on the propagation of singularities, which says that the wavefront set (position and direction of a singularity) of $e^{itP} u(x)$ is invariant under the Hamiltonian flow (viewed on the space $\mathbb{R}_t \times M_x$).

Edit: There lots of related results and the whole point of microlocal analysis is to relate the Hamiltonian dynamics to the PDE.

Answer 2

It goes something like this (I can't promise that what I've written below is completely correct. It is only to help you read the rigorous details in more definitive reference):

The initial observation is that the smoothness of a distribution $f$ on $\mathbb{R}^n$ can be measured using the decay rate of its Fourier transform $\hat{f}(\xi)$ as a function of $|\xi|$. Microlocal analysis means to localize this idea in the cotangent bundle.

The wavefront set $WF(u) \subset T^*M$ of a distribution $u$ on $M$ has the property that $$ WF_x(U) = WF(u)\cap T^*_xM $$ is a closed conic subset of $T^*_xM$.

First, it suffices to restrict to a compactly supported distribution $u$ on a coordinate chart $O \subset M$. We can therefore assume everything is on $\mathbb{R}^n$. Then $WF_x(u)$ at $x \in \mathbb{R}^n$ is a closed cone in $\mathbb{R}^n\backslash\{0\}$, which should be viewed as $T^*_x\mathbb{R}^n\backslash\{0\}$. In particular, $\xi_0 \notin WF_x(u)$ if and only if for any neighborhood $N$ of $x$ there exists $\chi \in C^\infty_0(N)$ such that the Fourier transform $\widehat{\chi u}$ decays rapidly (faster than polynomial decay) in a conical neighborhood of $\xi_0$.

Let $\sigma$ be the symbol of a real linear differential operator $P$ and $\Sigma = \sigma^{-1}(0)$ its characteristic variety. We say that $P$ is of real principal type if the Hamiltonian vector field of $\sigma$ restricted to $\Sigma$ is everywherer nonzero.

Hormander's original propagation of singularities theorem (which has many antecedents and by now many generalizations) says that if $P$ is a real differential operator of real principal type and $u$ is a distribution on $M$ such that $Pu \in C^\infty(M)$, then $WF(u) \subset \Sigma$ and is invariant under the null bicharacteristic flow, which is the Hamiltonian flow of $\sigma$ restricted to $\Sigma$.

Answer 3

I learned the following result from Sogge's book:

Let $P$ be a first order self-adjoint elliptic operator on $M$. Let $Q\in \Psi^{m}(M)$ be a classical $\Psi DO$ of order $m$. Then there exists a one parameter family of $\Psi DO$s: $t\rightarrow E(t)\in \Psi^{m}(M)$, depending smoothly on $t\in \mathbb{R}$, satisfying $$ [\partial_{t}-iP, E(t)]=0,E(0)=Q $$ and having for each $t\in \mathbb{R}$ the principal symbol $E_{0}(t,x,\xi)=q_{0}(\Psi_{t}(x,\xi))$ with $q_0(x,\xi)$ the principal symbol of $Q$ and $\Psi_{t}$ the Hamiltonian flow associated with $P$.

I do not know if there are more refined results directly concerning the linear equation $Qu=0$, though.