Finite Gap Jacobi Matrices, II. The Szegő Class

Abstract

Let e ⊂ R be a finite union of disjoint closed intervals. We study measures whose essential support is e and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to lim sup a_1...a_n/cap e_n >0 (this includes prior results of Widom and Peherstorfer-Yuditskii). Using Remling's extension of the Denisov-Rakhmanov theorem and an analysis of Jost functions, we provide a new proof of Szego asymptotics, including L^2 asymptotics on the spectrum. We make heavy use of the covering map formalism of Sodin-Yuditskii as presented in our first paper in this series.