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Semiclassical measures for the Schrödinger equation on the torus. (English) Zbl 1298.42028
Summary: In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant Lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the \(L^2\)-norm of a solution on any open subset of the torus controls the full \(L^2\)-norm.

42B37 Harmonic analysis and PDEs
35B40 Asymptotic behavior of solutions to PDEs
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