Semiclassical measure

Definition

Let $u_h$ be an $L^2$ -bounded family on a manifold as $h\to0$ . A semiclassical measure is a Radon measure $\mu$ on phase space obtained along a subsequence from

\langle \operatorname{Op}_h(a)u_h,u_h\rangle \longrightarrow \int a\,d\mu

for every compactly supported smooth symbol $a$ , where $\operatorname{Op}_h(a)$ is a fixed semiclassical quantization.

Dependence and invariance

The measure may depend on the chosen subsequence, but standard quantizations give the same limit. If $u_h$ solves a semiclassical eigenvalue equation, $\mu$ is supported on the corresponding characteristic energy surface and is invariant under the classical Hamiltonian flow.

Full support

For Laplace eigenfunctions on a compact hyperbolic surface, every semiclassical measure has full support. The stronger uniform statement is the eigenfunction mass lower bound.

References

Maciej Zworski, Semiclassical Analysis, AMS, 2012. Publisher record.