Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite-gap potentials and integrable geodesic equations on a 2-surface

S. V. Agapov, A. E. Mironov

TL;DR

The paper establishes that the one-dimensional Schrödinger equation can be viewed as the geodesic flow of a metric on a 2‑surface, with metrisability becoming explicit in the finite-gap case through the Baker--Akhiezer function. It connects spectral theory, theta-function representations, and differential geometry to produce explicit metric coefficients and geodesics in terms of finite-gap data. The main contribution is a constructive framework that expresses the metric, its curvature, and the corresponding geodesics via the Baker--Akhiezer function, thereby proving integrability of the geodesic equations for finite-gap potentials. This work deepens the link between integrable systems, spectral curves, and geometric flows, with concrete examples such as the Lame operator and a cuspidal degeneration illustrating the approach.

Abstract

In this paper we show that the one-dimensional Schrödinger equation can be viewed as the geodesic equation of a certain metric on a 2-surface. In case of the Schrödinger equation with a finite-gap potential, the metric and geodesics are explicitly found in terms of the Baker--Akhiezer function

Finite-gap potentials and integrable geodesic equations on a 2-surface

TL;DR

The paper establishes that the one-dimensional Schrödinger equation can be viewed as the geodesic flow of a metric on a 2‑surface, with metrisability becoming explicit in the finite-gap case through the Baker--Akhiezer function. It connects spectral theory, theta-function representations, and differential geometry to produce explicit metric coefficients and geodesics in terms of finite-gap data. The main contribution is a constructive framework that expresses the metric, its curvature, and the corresponding geodesics via the Baker--Akhiezer function, thereby proving integrability of the geodesic equations for finite-gap potentials. This work deepens the link between integrable systems, spectral curves, and geometric flows, with concrete examples such as the Lame operator and a cuspidal degeneration illustrating the approach.

Abstract

In this paper we show that the one-dimensional Schrödinger equation can be viewed as the geodesic equation of a certain metric on a 2-surface. In case of the Schrödinger equation with a finite-gap potential, the metric and geodesics are explicitly found in terms of the Baker--Akhiezer function
Paper Structure (5 sections, 2 theorems, 74 equations, 3 figures)

This paper contains 5 sections, 2 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction and the main result
  2. Finite-gap potentials and the Baker--Akhiezer function
  3. The Baker--Akhiezer function
  4. Finite-gap Schrödinger operators
  5. Proof of Theorem 1

Key Result

Theorem 1

The Schrödinger equation 6 is metrisable, and coefficients of the metric $ds^2$ have the form where $r_0$ is a constant. Here functions $s(x), l(x)$ satisfy the equations The Gaussian curvature $K$ of the metric $ds^2$ has the form

Figures (3)

  • Figure 1: $\beta_1=1,$$\beta_2=-1;$
  • Figure 2: $\beta_1=1,$$\beta_2=0;$
  • Figure 3: $\beta_1=1,$$\beta_2=2$

Theorems & Definitions (4)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Lemma 1