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Spectral Picard-Vessiot fields for Algebro-geometric Schrödinger operators

This work is a galoisian study of the spectral problem $LΨ=λΨ$, 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 $λ$ 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 $LΨ=λΨ$ analitically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.