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

Abstract

We analyse the structure of semiclassical and microlocal Wigner measures for\nsolutions to the linear Schr\\"{o}dinger equation on the disk, with Dirichlet\nboundary conditions.\n Our approach links the propagation of singularities beyond geometric optics\nwith the completely integrable nature of the billiard in the disk. We prove a\n"structure theorem", expressing the restriction of the Wigner measures on each\ninvariant torus in terms of {\\em second-microlocal measures}. They are obtained\nby performing a finer localization in phase space around each of these tori, at\nthe limit of the uncertainty principle, and are shown to propagate according to\nHeisenberg equations on the circle.\n Our construction yields as corollaries (a) that the disintegration of the\nWigner measures is absolutely continuous in the angular variable, which is an\nexpression of the dispersive properties of the equation; (b) an observability\ninequality, saying that the $L^2$-norm of a solution on any open subset\nintersecting the boundary (resp. the $L^2$-norm of the Neumann trace on any\nnonempty open set of the boundary) controls its full $L^2$-norm (resp.\n$H^1$-norm). These results show in particular that the energy of solutions\ncannot concentrate on periodic trajectories of the billiard flow other than the\nboundary.\n