Wigner measures and observability for the Schrodinger equation on the disk

• 成果类型: Article
• 署名作者: Anantharaman, Nalini; Leautaud, Matthieu; Macia, Fabricio
• 署名单位: Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universidad Politecnica de Madrid
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0658-4
• 发表日期: 2016
• 页码: 485-599
• 关键词: semiclassical measures; quantum limits; wave; CONTROLLABILITY; stabilization; Operators; DYNAMICS; FLOW; homogenization; systems

摘要

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrodinger 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 -norm of a solution on any open subset intersecting the boundary (resp. the -norm of the Neumann trace on any nonempty open set of the boundary) controls its full -norm (resp. -norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.