Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators

Abstract

This work is a Galoisian study of the spectral problem LΨ=λΨ, for an 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 λ as an algebraic variable over C, forcing the consideration of a new field of coefficients for L-λ, whose field of constants is the field C(Γ) of the spectral curve Γ. Since C(Γ) is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem over Γ, called “Spectral Picard–Vessiot field” of L-λ. An existence theorem is proved using differential algebra, allowing to recover classical Picard–Vessiot theory for each λ=λ 0 . For rational spectral curves, the appropriate algebraic setting is established to solve LΨ=λΨ analytically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.

Locations

• arXiv (Cornell University) - View - PDF
• Biblos-e Archivo (Universidad Autónoma de Madrid) - View - PDF
• French digital mathematics library (Numdam) - View - PDF
• Annales de l’institut Fourier - View - PDF