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The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator L in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator M to be in the centralizer of L. Whenever the centralizer equals the algebra generated by L and M , we call L, M a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order 4 in the first Weyl algebra. Moreover, for true rank r pairs, by means of differential subresultants, we effectively compute the fiber of the rank r spectral sheaf over their spectral curve.

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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 = λ 0 . 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.Résumé. -Ce travail est une étude Galoisienne du problème spectral LΨ = λΨ, 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 LΨ = λΨ de manière analytique et pour utiliser l'intégration symbolique. Nous illustrons nos résultats pour les solitons de Rosen-Morse.