Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators

Morales, Juan J.; Rueda, Sonia L.; Zurro, Maria-Angeles

doi:10.5802/aif.3425

Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators
[Corps spectraux de Picard–Vessiot pour les opérateurs de Schrödinger algébro-géométriques]

Morales, Juan J.¹ ; Rueda, Sonia L.² ; Zurro, Maria-Angeles ³

¹ Dpto. de Matemática Aplicada. E.T.S. Edificación. Universidad Politécnica de Madrid. Avda. Juan de Herrera 6. E-28040, Madrid (Spain)
² Dpto. de Matemática Aplicada. E.T.S. Arquitectura. Universidad Politécnica de Madrid. Avda. Juan de Herrera 4. E-28040, Madrid (Spain)
³ Dpto. de Matemáticas. Facultad de Ciencias. Universidad Autónoma de Madrid. Ciudad Universitaria de Cantoblanco. E-28049 Madrid (Spain)

Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1287-1324.

Résumé
Abstract

Ce travail est une étude Galoisienne du problème spectral $L Ψ = λ Ψ$ , pour les opérateurs différentiels algébro-géométriques du second ordre $L$ , avec des coefficients dans un corps différentiel, dont le corps de constantes $C$ est algébriquement clos et de caractéristique zéro. Notre approche considère le paramètre spectral $λ$ une variable algébrique sur $C$ , ce qui amène à considérer un nouveau corps de coefficients pour $L - λ$ , dont le corps de constantes est le champ $C (Γ)$ de la courbe spectrale $Γ$ . Puisque $C (Γ)$ n’est plus algébriquement clos, le besoin se fait sentir d’une nouvelle structure algébrique, générée par les solutions du problème spectral sur $Γ$ , appelée « Corps spectral de Picard–Vessiot » de $L - λ$ . On prouve un théorème d’existence en utilisant l’algèbre différentielle, permettant de retrouver la théorie classique de Picard–Vessiot pour chaque $λ = λ_{0}$ . Pour les courbes spectrales rationnelles, on établit le cadre algébrique approprié pour résoudre $L Ψ = λ Ψ$ de manière analytique et pour utiliser l’intégration symbolique. Nous illustrons nos résultats pour les solitons de Rosen-Morse.

This work is a Galoisian study of the spectral problem $L Ψ = λ Ψ$ , 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 $L Ψ = λ Ψ$ analytically and to use symbolic integration. We illustrate our results for Rosen-Morse solitons.

Reçu le : 2019-07-15
Révisé le : 2020-05-17
Accepté le : 2020-08-03
Première publication : 2021-12-08
Publié le : 2022-03-15

DOI : 10.5802/aif.3425

Classification : 12H05, 34M15
Keywords: Picard–Vessiot extension, Liouvillian extension, algebro-geometric operator, spectral curve
Mot clés : Extension de Picard–Vessiot, extension liouvillienne, opérateur algébro-géométrique, courbe spectrale

Affiliations des auteurs :

Morales, Juan J. ¹ ; Rueda, Sonia L. ² ; Zurro, Maria-Angeles ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Spectral {Picard{\textendash}Vessiot} fields for {Algebro-geometric} {Schr\"odinger} operators},
     journal = {Annales de l'Institut Fourier},
     pages = {1287--1324},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Morales, Juan J.; Rueda, Sonia L.; Zurro, Maria-Angeles. Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators. Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1287-1324. doi : 10.5802/aif.3425. https://aif.centre-mersenne.org/articles/10.5802/aif.3425/

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