Continuity of the spectrum of a field of self-adjoint operators

TL;DR

This paper establishes conditions under which the spectrum of a family of self-adjoint operators varies continuously with a parameter, including H"older continuity and bounds on gap closures.

Contribution

It provides necessary and sufficient conditions for spectral continuity and extends these to H"older continuity in complete metric spaces.

Findings

Spectral boundaries are continuous in the parameter.

Conditions for Vietoris and H"older continuity are characterized.

An upper bound for the size of closing spectral gaps is derived.

Abstract

Given a family of self-adjoint operators $(A_t)_{t\in T}$ indexed by a parameter $t$ in some topological space $T$, necessary and sufficient conditions are given for the spectrum $\sigma(A_t)$ to be Vietoris continuous with respect to $t$. Equivalently the boundaries and the gap edges are continuous in $t$. If $(T,d)$ is a complete metric space with metric $d$, these conditions are extended to guarantee H\"older continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.