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

TL;DR

This paper demonstrates that the one-dimensional Schrödinger equation with finite-gap potentials can be interpreted as a geodesic equation on a 2-surface, with explicit metrics and geodesics derived using Baker--Akhiezer functions.

Contribution

It establishes a novel geometric interpretation of finite-gap Schrödinger equations as geodesic flows on a 2-surface, providing explicit solutions.

Findings

Finite-gap potentials correspond to specific metrics on a 2-surface.

Explicit geodesic equations are derived using Baker--Akhiezer functions.

The geometric framework offers new insights into integrable systems.

Abstract

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