Growth of Sobolev norms for unbounded perturbations of the Schr\"odinger equation on flat tori

TL;DR

This paper establishes a polynomial-in-time bound on the growth of Sobolev norms for Schrödinger equations with unbounded, time-dependent perturbations on flat tori, advancing understanding of long-term behavior in such quantum systems.

Contribution

It provides the first polynomial bound on Sobolev norm growth for unbounded perturbations of the Schrödinger equation on flat tori.

Findings

Sobolev norms grow at most polynomially in time

Bound of $ extless t angle^ extvarepsilon$ established

Results apply to unbounded, time-dependent perturbations

Abstract

We prove a $\langle t\rangle^\varepsilon$ bound on the growth of Sobolev norms for unbounded time dependent perturbations of the Laplacian on flat tori.