EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY

By Barry Simon. 2007

113 citations recorded by the source corpus.

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--Szegő theory.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4c9357219e75c93cc66d735330ee848258d32dc9

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

The Christoffel-Darboux Kernel (2008, background): “For the following, see [91, 85]: Theorem 12.”
Schrödinger operators with dynamically defined potentials (2014, background): “Using this interpretation, the following result of Simon [195] (which is essentially already in [199]) is not too difficult to deduce.”
Reflectionless Herglotz Functions and Jacobi Matrices (2009, unknown): “See, for example, [ 15 ]; the book [19] has a systematic exposition of similar methods in a slightly different setting.”
The Analytic Theory of Matrix Orthogonal Polynomials (2007, methodology): “The result below can be proven using polynomial mappings à la Totik [188] or Jost solutions à la Simon [174].)”