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Published May 12, 2017 | Submitted
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Equilibrium measures and capacities in spectral theory

Abstract

This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schr¨odinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete–Szegő theory.

Additional Information

August 23, 2007. (Submitted on 16 Nov 2007) Supported in part by NSF grants DMS-0140592 and DMS-0652919 and U.S.–Israel Binational Science Foundation (BSF) Grant No.2002068.

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August 19, 2023
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