Finite Gap Jacobi Matrices, I. The Isospectral Torus

Abstract

Let e ⊂ R be a finite union of disjoint closed intervals. In the study of orthogonal polynomials on the real line with measures whose essential support is 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.