AF embeddings and the numerical computation of spectra in irrational rotation algebras

Abstract

The spectral analysis of discretized one-dimensional Schr\"{o}dinger operators is a very difficult problem which has been studied by numerous mathematicians. A natural problem at the interface of numerical analysis and operator theory is that of finding finite dimensional matrices whose eigenvalues approximate the spectrum of an infinite dimensional operator. In this note we observe that the seminal work of Pimsner-Voiculescu on AF embeddings of irrational rotation algebras provides a nice answer to the finite dimensional spectral approximation problem for a broad class of operators including the quasiperiodic case of the Schr\"{o}dinger operators mentioned above. Indeed, the theory of continued fractions not only provides good matrix models for spectral computations (i.e. the Pimsner-Voiculescu construction) but also yields {\em sharp} rates of convergence for spectral approximations of operators in irrational rotation algebras.