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...

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Bibliographic Details
Published in:Numerical functional analysis and optimization 2006-09, Vol.27 (5-6), p.517-528
Main Author: Brown, Nathanial P.
Format: Article
Language:English
Subjects:
AF embedding
Approximation
Spectrum
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Summary: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.
ISSN:0163-0563
1532-2467
DOI:10.1080/01630560600790785