AF embeddings and the numerical computation of spectra in irrational rotation algebras
Nathanial P. Brown
TL;DR
The work shows that spectral quantities of operators in AF algebras, notably irrational rotation algebras, can be effectively approximated by finite-dimensional matrices via the Pimsner-Voiculescu construction, with sharp convergence rates tied to continued fraction inputs. By exploiting convergents p_n/q_n, the paper builds explicit matrix models h_{θn} that converge to the target operator Hθ, providing explicit two-sided pseudospectral and, in the normal case, spectral distance bounds. It also delivers one-sided rate results for spectral convergence as a function of θ-θ′ using Haagerup–Rørdam continuity, offering implementable guarantees for a broad class of quasiperiodic operators, including discretized Schrödinger operators with polynomial potential. The findings illuminate both the practical computability of spectra via finite models and the intrinsic limits posed by irrational approximation properties.
Abstract
The spectral analysis of discretized one-dimensional Schrödinger 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 Schrödinger 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.