Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators
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 a...
| Autores: | , , |
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| Formato: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Recursos: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/709764 |
| Acesso em linha: | http://hdl.handle.net/10486/709764 https://dx.doi.org/10.5802/aif.3425 |
| Access Level: | acceso abierto |
| Palavra-chave: | Picard–Vessiot extension Liouvillian extension algebro-geometric operator spectral curve Matemáticas |
| Resumo: | 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 = λ0. 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 |
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