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Wigner measures and observability for the Schrödinger equation on the disk. (English) Zbl 1354.35125

The authors study the dynamical properties of the two-dimensional time-dependent Schrödinger equation \({{1} \over {i}} {{\partial u} \over {\partial t}}(x,y,t) = (-{{1} \over {2}} \Delta + V(x,y,t))u(x,y,t)\) in the unit disc with homogeneous Dirichlet boundary conditions; \(V\) is a smooth real valued potential. In order to perform this study one has to introduce the dual variables, momentum and energy. Thus the authors were brought to investigate the regularity und localization properties of the Wigner measures on the phase space.
The main results are two theorems, which provide a detailed structure of the microlocal and semiclassical Wigner measures, defined in the paper, associated to sequences of solutions of the Schrödinger equation.
As consequences one obtains: two continuation properties for the Schrödinger equation which state the controlablity and the interior and boundary observability of the equation and a result on the dispersive character of the Schrödinger equation.
Similar results were obtained for the Schrödinger equation on flat tori in [N. Anantharaman et al., Am. J. Math. 137, No. 3, 577–638 (2015; Zbl 1319.35207)] and in [N. Anantharaman and F. Macià, J. Eur. Math. Soc. (JEMS) 16, No. 6, 1253–1288 (2014; Zbl 1298.42028)].

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
93B07 Observability
35Q40 PDEs in connection with quantum mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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