# American Institute of Mathematical Sciences

November  2007, 1(4): 713-772. doi: 10.3934/ipi.2007.1.713

## Equilibrium measures and capacities in spectral theory

 1 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125

Received  August 2007 Published  October 2007

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 Schrödinger 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.
Citation: Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
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