2022
DOI: 10.5802/aif.3425
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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 … Show more

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Cited by 5 publications
(12 citation statements)
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“…be a Boussinesq operator. We define the ideal associated to L to be the ideal in C[λ, µ, γ] generated by the set {f 1 , f 2 , f 3 }, with f i defined in (14). We denote this ideal by I(L).…”
Section: Differential Resultants and Spectral Curvesmentioning
confidence: 99%
See 3 more Smart Citations
“…be a Boussinesq operator. We define the ideal associated to L to be the ideal in C[λ, µ, γ] generated by the set {f 1 , f 2 , f 3 }, with f i defined in (14). We denote this ideal by I(L).…”
Section: Differential Resultants and Spectral Curvesmentioning
confidence: 99%
“…The present work is the natural continuation in a program dedicated to the factorization of rank 1 algebro-geometric differential operators, that was already successful in the order 2 case, [13]. Our ultimate goal is an effective approach to the direct spectral problem and the development of the appropriate spectral Picard-Vessiot fields containing all the solutions of the operator L − λ. Spectral Picard-Vessiot fields were studied for Schrödinger operators in [14]. The development of algorithms for their computation is part of an ongoing project that focuses on the study of the centralizers of operators with potentials verifying one of Gel'fand-Dickii integrable hierarchies [11].…”
Section: Discussionmentioning
confidence: 99%
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“…points out the interest of studying the centralizer C(L) of L in R ℓ [D]. The results of K. Goodearl in [11] about centralizers of ODOs have proven to be important for effective computational approaches in [31,25,26]. In this sense, a generalization to the case of MODOs of Goodearl's results would give an effective description of C(L) but we do not achieve such description in this paper.…”
Section: Matrix Coefficient Odosmentioning
confidence: 87%