C*-algebras and numerical linear algebra

TL;DR

This paper explores the spectral properties of self-adjoint operators through their finite sections, revealing how eigenvalues relate to the essential spectrum and connecting numerical analysis with C*-algebra theory.

Contribution

It establishes a link between eigenvalue distributions of finite sections and the essential spectrum using C*-algebra frameworks, especially for quantum Hamiltonians.

Findings

Eigenvalues accumulate near the essential spectrum.

Averages of eigenvalues converge to a measure on the essential spectrum.

Results apply to discretized quantum Hamiltonians in one dimension.

Abstract

We consider problems associated with the computation of spectra of self-adjoint operators in terms of the eigenvalue distributions of their n x n sections. Under rather general circumstances, we show how these eigenvalues accumulate near points of the essential spectrum of the given operator, and we prove that their averages converge to a measure concentrated precisely on the essential spectrum. In the primary cases of interest, namely the discretized Hamiltonians of one-dimensional quantum systems, this limiting measure is associated with a tracial state on a certain simple C*-algebra. These results have led us to conclude that one must view this kind of numerical analysis in the context of C*-algebras.