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Journal Articles Inventiones Mathematicae Year : 2016

Wigner measures and observability for the Schrödinger equation on the disk

Abstract

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 two-microlocal measures, propagating according to Schrödinger 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 different from the boundary.
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Dates and versions

hal-00999330 , version 1 (03-06-2014)

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Nalini Anantharaman, Matthieu Léautaud, Fabricio Macià. Wigner measures and observability for the Schrödinger equation on the disk. Inventiones Mathematicae, 2016, 206 (2), pp.485-599. ⟨10.1007/s00222-016-0658-4⟩. ⟨hal-00999330⟩
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