Long-time dynamics of the perturbed Schr\"odinger equation on negatively curved surfaces

TL;DR

This paper studies the long-time behavior of solutions to the perturbed Schrödinger equation on negatively curved surfaces, showing equidistribution at logarithmic times and analyzing quantum Loschmidt echo properties.

Contribution

It demonstrates equidistribution of solutions at logarithmic times for perturbations on negatively curved surfaces and explores quantum Loschmidt echo behavior beyond Ehrenfest time.

Findings

Solutions become equidistributed at logarithmic times

Quantum Loschmidt echo properties are characterized

Results apply to initial data in small spectral windows

Abstract

We consider perturbations of the semiclassical Schr{\"o}dinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.