Globally integrable quantum systems and their perturbations

TL;DR

This paper introduces the concept of globally integrable quantum systems, explores their spectral properties, and analyzes the stability and growth of eigenvalues and Sobolev norms under various perturbations.

Contribution

It formalizes the notion of globally integrable quantum systems and provides new spectral and stability results for their perturbations.

Findings

Eigenvalues remain stable under relatively bounded perturbations

Sobolev norms grow controllably under unbounded linear perturbations

Small nonlinear perturbations do not destabilize the system

Abstract

In this paper we present the notion of globally integrable quantum system that we introduced in [BL22]: we motivate it using the spectral theory of pseudodifferential operators and then we give some results on linear and nonlinear perturbations of a globally integrable quantum system. In particular, we give a spectral result ensuring stability of most of its eigenvalues under relatively bounded perturbations, and two results controlling the growth of Sobolev norms when it is subject either to linear unbounded time dependent perturbations or a small nonlinear Hamiltonian nonlinear perturbation.