Perturbation of the semiclassical Schr\"odinger equation on negatively curved surfaces

TL;DR

This paper studies how small perturbations affect the quantum evolution of the semiclassical Schr"odinger equation on negatively curved surfaces, showing that solutions tend to become uniformly distributed in the semiclassical limit.

Contribution

It demonstrates that, under small perturbations, solutions to the semiclassical Schr"odinger equation on negatively curved surfaces become equidistributed, extending understanding of quantum chaos and stability.

Findings

Solutions become equidistributed under typical small perturbations

Results hold below the Ehrenfest time in the semiclassical limit

Applicable to initial data microlocalized on the unit cotangent bundle

Abstract

We consider the semiclassical Schr\"odinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schr\"odinger equation, and we prove that, in the semiclassical limit and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.