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

P. A. Leonchik, G. S. Mauleshova, A. E. Mironov

TL;DR

The paper develops a two-dimensional analogue of the finite-gap correspondence by linking two-dimensional finite-gap Schrödinger operators at a fixed energy to two-dimensional integrable difference operators. By constructing two-parameter difference operators $L_{\varepsilon,\delta}$ with kernels containing a two-point Baker–Akhiezer function and expressing operator coefficients via theta functions, the authors show that, as $(\varepsilon,\delta) \to (0,0)$, these discrete operators converge to the 2D Schrödinger operator $H = \partial_z\partial_{\bar z} + A(z,\bar z)\partial_{\bar z} + u(z,\bar z)$ while preserving the spectral curve. The work extends the KN rank-one framework from 1D to 2D, providing explicit theta-function constructions for both discrete BA functions and the associated operators, and demonstrates a rigorous discrete-to-continuum limit for integrable 2D systems. This advances the understanding of how discrete integrable systems approximate their continuous finite-gap counterparts in higher dimensions and informs geometric applications and spectral theory in 2D.

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.

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

TL;DR

The paper develops a two-dimensional analogue of the finite-gap correspondence by linking two-dimensional finite-gap Schrödinger operators at a fixed energy to two-dimensional integrable difference operators. By constructing two-parameter difference operators with kernels containing a two-point Baker–Akhiezer function and expressing operator coefficients via theta functions, the authors show that, as , these discrete operators converge to the 2D Schrödinger operator while preserving the spectral curve. The work extends the KN rank-one framework from 1D to 2D, providing explicit theta-function constructions for both discrete BA functions and the associated operators, and demonstrates a rigorous discrete-to-continuum limit for integrable 2D systems. This advances the understanding of how discrete integrable systems approximate their continuous finite-gap counterparts in higher dimensions and informs geometric applications and spectral theory in 2D.

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.

Paper Structure

This paper contains 6 sections, 3 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Two-Dimensional Finite-Gap Schrödinger Operators at One Energy Level
  3. One-Dimensional Finite-Gap Schrödinger Operators
  4. Two-Point Baker--Akhiezer Function
  5. Two-Dimensional Difference Operators
  6. Two-Dimensional Difference Operators

Key Result

Theorem 1

(LM) There exists a unique meromorphic function $\psi(n,m,P)$ on $\Gamma$, $n,m\in\mathbb{Z}$, $P\in\Gamma$, satisfying the following conditions: 1. The zero and pole divisor of $\psi$ has the form where $\gamma_{n,m} = \gamma_1(n,m) + \ldots + \gamma_g(n,m)$ is some divisor on $\Gamma$, with $\gamma_{0,0} = \gamma$. 2. In a neighbourhood of $p$, the function $\psi(n,m,P)$ has the form where $

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3