Semiclassical measures for inhomogeneous Schrödinger equations on tori

Abstract

The purpose of this note is to investigate the high-frequency behavior of solutions to linear Schrödinger equations. More precisely, Bourgain (1997) and Anantharaman and Macià (2011) proved that any weak-$\ast$ limit of the square density of solutions to the time-dependent homogeneous Schrödinger equation is absolutely continuous with respect to the Lebesgue measure on $ℝ\times {\mathbb{T}}^d$. The contribution of this article is that the same result automatically holds for nonhomogeneous Schrödinger equations, which allows for abstract potential type perturbations of the Laplace operator.