Matthieu Léautaud

Wigner measures and observability for the Schrödinger equation on the disk

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger 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 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

Energy decay for a locally undamped wave equation

L^2

Uniform observability estimates for linear waves

L2-norm of a solution on any open subset intersecting the boundary (resp. the

Delocalization of quasimodes on the disk

L^2

Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller

L2-norm of the Neumann trace on any nonempty open set of the boundary) controls its full