Finite Gap Jacobi Matrices, II. The Szegő Class

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frak{e}\subset\mathbb{R}$\end{document} be a finite union of disjoint closed intervals. We study measures whose essential support is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\frak{e}}$\end{document} and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup\frac{a_1\cdots a_n}{\mathrm{cap}(\frak{e})^n}>0$$\end{document} (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 L2 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.