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Bulletin de la SMF - Titles - 145 (2017) 47-95

Titles < 145

Quadratic operator pencils associated with the conservative Camassa-Holm flow
Jonathan Eckhardt & Aleksey Kostenko
Bulletin de la SMF 145, fascicule 1 (2017), 47-95

Télécharger cet article : Fichier PDF

Résumé :
Pinceaux quadratiques d'opérateurs associés au flot Camassa-Holm conservateur
Nous discutons la théorie spectrale directe et inverse pour un problème tapez Sturm-Liouville avec une dépendance quadratique du paramètre valeur propre, -f'' + 14 f = z f + z^2 v f, qui se pose le problème isospectral pour le flot Camassa-Holm conservateur. Afin d'être capable de traiter des coefficients plutôt irréguliers (qui est, quand est une mesure de Borel de valeur réelle sur R et est une mesure de Borel non-négative sur R), nous employons une nouvelle approche pour étudier ce problème spectral. En particulier, nous fournissons des résultats auto-adjointness de base pour des réalisations dans des espaces de Hilbert appropriés, développons théorie Weyl-Titchmarsh (singulier) et prouvons plusieurs théorèmes de base d'unicité inverse pour ce problème spectral.

Mots-clefs : Problème de Sturm-Liouville, pinceaux quadratiques d’opérateurs, théorie spectrale (inverse).

Abstract:
We discuss direct and inverse spectral theory for a Sturm-Liouville type problem with a quadratic dependence on the eigenvalue parameter, -f'' + 14 f = z f + z^2 v f, which arises as the isospectral problem for the conservative Camassa-Holm flow. In order to be able to treat rather irregular coefficients (that is, when is a real-valued Borel measure on R and v is a non-negative Borel measure on R), we employ a novel approach to study this spectral problem. In particular, we provide basic self-adjointness results for realizations in suitable Hilbert spaces, develop (singular) Weyl-Titchmarsh theory and prove several basic inverse uniqueness theorems for this spectral problem.

Keywords: Sturm-Liouville problems, quadratic operator pencils, (inverse) spectral theory.

Class. math. : 34L05, 34B07; 34B20, 37K15.


ISSN : 0037-9484
DOI : 10.24033/bsmf.2731
Publié avec le concours de : Centre National de la Recherche Scientifique

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