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

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.