Finite Gap Jacobi Matrices, I. The Isospectral Torus

Abstract

Let $\mathfrak{e}\subset \mathbb{R}$ be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is $\mathfrak{e}$, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.

68 pages, 4 figures

References in corpus (3)

• Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum
• Eigenvalue bounds in the gaps of Schrodinger operators and Jacobi matrices
• Finite Gap Jacobi Matrices: An Announcement