AF Embeddings and the Numerical Computation of Spectra in Irrational Rotation Algebras

Abstract

A natural problem at the interface of operator theory and numerical analysis is that of finding a (finite dimensional) matrix whose eigenvalues approximate the spectrum of a given (infinite dimensional) operator. It is well-known that classical work of Pimsner and Voiculescu produces explicit matrix models for an interesting class of nontrivial examples (e.g., many discretized one-dimensional Schrödinger operators). In this paper, we observe that the spectra of their models (often) converge in the strongest possible sense—in the Hausdorff metric—and demonstrate that the rate of convergence is, in general, best possible.