Two-microlocal regularity of quasimodes on the torus

TL;DR

This paper investigates the regularity and concentration properties of quasimodes for perturbed Schrödinger equations on 2D tori, revealing that accurate quasimodes concentrate only at critical points of averaged potentials.

Contribution

It introduces a second microlocal analysis framework to study quasimodes on tori, providing new insights into their concentration behavior under perturbations.

Findings

Quasimodes concentrate only at critical points of the averaged potential.

Second microlocalization effectively analyzes concentration at new scales.

Results apply to both stationary and time-dependent solutions.

Abstract

We study the regularity of stationary and time-dependent solutions to strong perturbations of the free Schr\"odinger equation on two-dimensional flat tori. This is achieved by performing a second microlocalization related to the size of the perturbation and by analysing concentration and nonconcentration properties at this new scale. In particular, we show that sufficiently accurate quasimodes can only concentrate on the set of critical points of the average of the potential along geodesics.