Hamiltonians with purely discrete spectrum

TL;DR

This paper explores conditions under which certain self-adjoint operators on L^2 spaces have no essential spectrum, providing general criteria and examples for locally compact abelian groups like R^n.

Contribution

It introduces new criteria for the absence of essential spectrum in self-adjoint operators on L^2 spaces, with specific examples for abelian groups.

Findings

Criteria for empty essential spectrum established

General results for locally compact abelian groups

Examples provided for R^n

Abstract

We discuss criteria for a self-adjoint operator on L^2(X) to have empty essential spectrum. We state a general result for the case of a locally compact abelian group X and give examples for X=R^n.