Spectral Picard-Vessiot fields for Algebro-geometric Schrödinger operators

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 spectr...

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Bibliographic Details
Published inarXiv.org
Main Authors Morales-Ruiz, Juan J, Rueda, Sonia L, Maria-Angeles Zurro
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.02.2021
Subjects
Algebra
Closed form solutions
Differential geometry
Exact solutions
Operators (mathematics)
Parameterization
Resultants
Spectra
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Summary: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.
ISSN:2331-8422