Spectral Picard-Vessiot fields for Algebro-geometric Schrödinger operators

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.

To appear in Annales de l'Institut Fourier