Spectral Picard-Vessiot fields for Algebro-geometric Schr\"odinger operators

TL;DR

This paper develops a Galoisian framework for spectral problems of algebro-geometric Schrödinger operators, introducing Spectral Picard-Vessiot fields to handle the spectral parameter over spectral curves, and applies it to Rosen-Morse solitons.

Contribution

It introduces Spectral Picard-Vessiot fields for algebro-geometric Schrödinger operators, extending classical Picard-Vessiot theory to spectral curves with new algebraic structures.

Findings

Established existence of Spectral Picard-Vessiot fields using differential algebra.

Provided an algebraic setting for solving spectral problems analytically.

Applied the theory to Rosen-Morse solitons.

Abstract

This work is a galoisian study of the spectral problem $L\Psi=\lambda\Psi$, for algebro-geometric second order differential operators $L$, with coefficients in a differential field, whose field of constants $C$ is algebraically closed and of characteristic zero. Our approach regards the spectral parameter $\lambda$ an algebraic variable over $C$, forcing the consideration of a new field of coefficients for $L-\lambda$, whose field of constants is the field $C(\Gamma)$ of the spectral curve $\Gamma$. Since $C(\Gamma)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem over $\Gamma$, called "Spectral Picard-Vessiot field" of $L-\lambda$. An existence theorem is proved using differential algebra, allowing to recover classical Picard-Vessiot theory for each $ \lambda = \lambda_0 $. For rational spectral curves, the appropriate algebraic setting is established to solve $L\Psi=\lambda\Psi$ analitically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.