Perturbation Theory of Schr\"odinger Operators in Infinitely Many Coupling Parameters

TL;DR

This paper investigates how Hamilton operators with infinitely many coupling parameters behave, focusing on their spectral properties, stability, and analyticity of eigenvalues, motivated by models of quantum particles in disordered media.

Contribution

It develops a framework for analyzing Schrödinger operators with infinitely many parameters, including stability of selfadjointness and eigenvalue analyticity in Banach space settings.

Findings

Operators remain selfadjoint under certain perturbations.

Eigenvalues depend analytically on parameters within specified classes.

Framework applies to models of quantum particles in disordered media.

Abstract

In this paper we study the behavior of Hamilton operators and their spectra which depend on infinitely many coupling parameters or, more generally, parameters taking values in some Banach space. One of the physical models which motivate this framework is a quantum particle moving in a more or less disordered medium. One may however also envisage other scenarios where operators are allowed to depend on interaction terms in a manner we are going to discuss below. The central idea is to vary the occurring infinitely many perturbing potentials independently. As a side aspect this then leads naturally to the analysis of a couple of interesting questions of a more or less purely mathematical flavor which belong to the field of infinite dimensional holomorphy or holomorphy in Banach spaces. In this general setting we study in particular the stability of selfadjointness of the operators under discussion and the analyticity of eigenvalues under the condition that the perturbing potentials belong to certain classes.