Localizations at infinity and essential spectrum of quantum Hamiltonians: I. General theory

TL;DR

This paper develops a general theoretical framework for understanding the essential spectrum of quantum Hamiltonians by analyzing their behavior at infinity within an abstract abelian group setting.

Contribution

It introduces a canonical representation of the essential spectrum based on limits at infinity of translated operators, applicable to a broad class of self-adjoint operators.

Findings

Essential spectrum determined by behavior at infinity

Canonical representation in terms of spectra of limits at infinity

Applicable to operators on arbitrary abelian locally compact groups

Abstract

We isolate a large class of self-adjoint operators H whose essential spectrum is determined by their behavior at large x and we give a canonical representation of their essential spectrum in terms of spectra of limits at infinity of translations of H. The configuration space is an arbitrary abelian locally compact not compact group.