Finite Gap Jacobi Matrices, Ii. the Szegő Class

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 a1 · · · an cap(e) > 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 Szegő asymptotics, including L asymptotics on the spectrum. We use heavily the covering map formalism of Sodin–Yuditskii as presented in our first paper in this series.