Semi-classical measures for inhomogeneous Schr\"odinger equations on tori

TL;DR

This paper extends the understanding of high frequency limits of solutions to inhomogeneous Schrödinger equations on tori, showing that their density limits remain absolutely continuous, similar to the homogeneous case, even with potential perturbations.

Contribution

It demonstrates that the absolute continuity of weak-* limits of solution densities holds for inhomogeneous Schrödinger equations with potential perturbations, generalizing previous homogeneous results.

Findings

Weak-* limits are absolutely continuous for inhomogeneous equations.

Results apply to Schrödinger equations with potential perturbations.

Extends previous homogeneous case findings.

Abstract

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