Equilibrium measures and capacities in spectral theory

TL;DR

This paper reviews potential theory's role in spectral analysis of orthogonal polynomials, highlighting the Stahl-Totik theory and linking to ergodic Schrödinger operators, with new results on spectral measure support.

Contribution

It introduces a new general result showing spectral measures are supported on sets of zero Hausdorff dimension in certain spectral regions.

Findings

Spectral measure support can have zero Hausdorff dimension.

Links established between potential theory and spectral properties of operators.

New conjectures and research directions proposed.

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 Schrodinger 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-Szego theory.