Two-dimensional discrete operators and rational functions on algebraic curves

TL;DR

This paper explores the relationship between finite-gap two-dimensional Schrödinger operators and discrete operators, identifying spectral data for integrable discrete operators and linking them to algebraic spectral curves, especially for genus one.

Contribution

It introduces a new class of two-dimensional integrable discrete operators with eigenfunctions on algebraic spectral curves and connects them to finite-gap Schrödinger operators in the genus one case.

Findings

Spectral data for new integrable discrete operators identified

Eigenfunctions parameterized by algebraic spectral curves

Finite-gap Schrödinger operators obtained as limits of discrete operators

Abstract

In this paper we study a connection between finite-gap on one energy level two-dimensional Schrodinger operators and two-dimensional discrete operators. We find spectral data for a new class of two-dimensional integrable discrete operators. These operators have eigenfunctions on zero level energy parameterized by points of algebraic spectral curves. In the case of genus one spectral curves we show that the finite-gap Schrodinger operators can be obtained as a limit of the discrete operators.