Two-Dimensional Finite-Gap Schrodinger Operators as Limits of Two-Dimensional Integrable Difference Operators

TL;DR

This paper explores how two-dimensional finite-gap Schrödinger operators can be derived as limits of discrete difference operators, providing insights into their spectral properties and the transition from discrete to continuous models.

Contribution

It introduces a method to obtain two-dimensional finite-gap Schrödinger operators from difference operators through a limiting process, extending the understanding of their spectral structure.

Findings

Finite-gap Schrödinger operators can be realized as limits of difference operators.

Eigenfunctions at zero energy are rational functions on spectral curves.

The approach bridges discrete and continuous two-dimensional operators.

Abstract

In this paper we study two-dimensional discrete operators whose eigenfunctions at zero energy level are given by rational functions on spectral curves. We extend discrete operators to difference operators and show that two-dimensional finite-gap Schrodinger operators at fixed energy level can be obtained from difference operators by passage to the limit.