Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two-dimensional discrete operators and rational functions on algebraic curves

Polina A. Leonchik, Andrey E. Mironov

TL;DR

The paper builds a bridge between finite-gap 2D Schrödinger operators at a single energy level and a new class of 2D integrable discrete operators by introducing spectral data on an algebraic curve Γ and a corresponding Baker–Akhiezer function. It proves a reconstruction framework (Theorem 1) that associates to spectral data a unique discrete operator L and a family of eigenfunctions ψ(n,m,P) parameterized by Γ, with Lψ = 0 and ψ exhibiting a fixed divisor structure. In the genus-one case, the continuous operator H emerges as a limit of the discrete operator L, and a difference operator L_{ε,δ} is shown to converge to H as ε,δ → 0 using elliptic-function data, establishing a discrete-to-continuous correspondence for algebro-geometric 2D integrable systems. These results extend the finite-gap paradigm to discrete 2D settings, clarifying how theta-function and Weierstrass-function data encode the spectral geometry and enabling potential applications in spectral theory and integrable systems.

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.

Two-dimensional discrete operators and rational functions on algebraic curves

TL;DR

The paper builds a bridge between finite-gap 2D Schrödinger operators at a single energy level and a new class of 2D integrable discrete operators by introducing spectral data on an algebraic curve Γ and a corresponding Baker–Akhiezer function. It proves a reconstruction framework (Theorem 1) that associates to spectral data a unique discrete operator L and a family of eigenfunctions ψ(n,m,P) parameterized by Γ, with Lψ = 0 and ψ exhibiting a fixed divisor structure. In the genus-one case, the continuous operator H emerges as a limit of the discrete operator L, and a difference operator L_{ε,δ} is shown to converge to H as ε,δ → 0 using elliptic-function data, establishing a discrete-to-continuous correspondence for algebro-geometric 2D integrable systems. These results extend the finite-gap paradigm to discrete 2D settings, clarifying how theta-function and Weierstrass-function data encode the spectral geometry and enabling potential applications in spectral theory and integrable systems.

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.
Paper Structure (4 sections, 1 theorem, 65 equations)

This paper contains 4 sections, 1 theorem, 65 equations.

Table of Contents

  1. Introduction and main results
  2. Finite-gap Schrödinger operators
  3. Discrete analogue of Schrödinger operator
  4. Difference analogue of Schödinger operator

Key Result

Theorem 1

There exists a unique meromorphic function $\psi(n,m, P)$ on $\Gamma$, $n,m \in \mathbb Z, P \in \Gamma$ with the following properties. Moreover, the function $\psi(n,m,P)$ satisfies the following equation where $a_{n,m}, b_{n,m}, v_{n,m}$ are some coefficients.

Theorems & Definitions (1)

  • Theorem 1