Semi-classical measures for inhomogeneous Schrödinger equations on tori

2012/09/17 by Nicolas Burq, Burq, Nicolas · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.AP #math.FA #msc:35LXX

paper · pdf · doi:10.48550/arxiv.1209.3739

Corrected a few typos, added, as an illustration, an appplication to a nonlinear equation

arxiv created 2012/12/20 · arxiv updated 2012/12/21

Abstract

The purpose of this note is to investigate the high frequency behaviour of solutions to linear Schrödinger equations. More precisely, Bourgain and Anantharaman-Macia proved that any weak-* limit of the square density of solutions to the time dependent homogeneous Schrödinger equation is absolutely continuous with respect to the Lebesgue measure on R× T^d. Our contribution is that the same result automatically holds for non homogeneous Schrödinger equations, which allows for abstract potential type perturbations of the Laplace operator.

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• Observability of Schrödinger propagators on tori in rough settings

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