Finite Gap Jacobi Matrices, I. The Isospectral Torus

TL;DR

This paper studies the isospectral torus associated with finite unions of intervals, using a covering map approach to analyze properties, bounds, and Jost functions expressed via theta functions, aiding in the study of perturbations.

Contribution

It introduces a covering map formalism to define and analyze the isospectral torus for finite gap Jacobi matrices, including explicit Jost function expressions.

Findings

Expression of Jost functions in terms of theta functions

Properties and bounds for the isospectral torus

Framework for studying perturbations of finite gap Jacobi matrices

Abstract

Let $\frak{e}\subset\mathbb{R}$ be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is $\frak{e}$, 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.