Meromorphic continuations of finite gap Herglotz functions and periodic Jacobi matrices

TL;DR

This paper characterizes when a Herglotz function can be the Borel transform of a spectral measure for perturbed periodic Jacobi matrices, using meromorphic continuation and zero-pole structure on a Riemann surface.

Contribution

It provides a necessary and sufficient condition for such Herglotz functions to correspond to spectral measures of exponentially decaying or eventually periodic Jacobi matrices, generalizing previous results.

Findings

Characterization of spectral measures via meromorphic continuation.

Extension of results to eventually periodic Jacobi matrices.

Generalization of earlier work on free Jacobi matrices.

Abstract

We find a necessary and sufficient condition for a Herglotz function $m$ to be the Borel transform of the spectral measure of an exponentially decaying perturbation of a periodic Jacobi matrix. The condition is in terms of meromorphic continuation of $m$ to a natural Riemann surface and the structure of its zeros and poles. The analogous result is also established for the Borel transform of the spectral measure of eventually periodic Jacobi matrices. This paper generalizes the corresponding result from [17] for exponentially decaying perturbations of the free Jacobi matrix.