Wigner measures and observability for the Schr\"odinger equation on the disk

TL;DR

This paper studies the behavior of solutions to the Schrödinger equation on a disk, revealing how their phase space measures propagate and establishing observability inequalities that prevent energy concentration on certain trajectories.

Contribution

It introduces a structure theorem linking Wigner measures to second-microlocal measures and proves new observability inequalities for the Schrödinger equation on the disk.

Findings

Wigner measures disintegrate absolutely continuously in the angular variable.

Solutions' energy cannot concentrate on non-boundary periodic trajectories.

The propagation of second-microlocal measures follows Heisenberg equations on the circle.

Abstract

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schr\"{o}dinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a "structure theorem", expressing the restriction of the Wigner measures on each invariant torus in terms of {\em second-microlocal measures}. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the $L^2$-norm of a solution on any open subset intersecting the boundary (resp. the $L^2$-norm of the Neumann trace on any nonempty open set of the boundary) controls its full $L^2$-norm (resp. $H^1$-norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.