Quadratic operator pencils associated with the conservative Camassa-Holm flow

Eckhardt, Jonathan; Kostenko, Aleksey

doi:10.24033/bsmf.2731

Quadratic operator pencils associated with the conservative Camassa-Holm flow
[Pinceaux quadratiques d’opérateurs associés au flot Camassa-Holm conservateur]

Eckhardt, Jonathan ¹ ; Kostenko, Aleksey ²

¹ School of Computer Science & Informatics, Cardiff University, Queen’s Buildings, 5 The Parade, Roath, Cardiff CF24 3AA, Wales, UK
² Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 1, pp. 47-95

Abstract (VO)
Résumé

We discuss direct and inverse spectral theory for a Sturm-Liouville type problem with a quadratic dependence on the eigenvalue parameter,

- f^{''} + \frac{1}{4} f = z ω f + z^{2} υ 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 $ω$ is a real-valued Borel measure on $ℝ$ and $υ$ 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.

Nous discutons la théorie spectrale directe et inverse pour un problème tapez Sturm-Liouville avec une dépendance quadratique du paramètre valeur propre,

- f^{''} + \frac{1}{4} f = z ω f + z^{2} υ 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 $ω$ est une mesure de Borel de valeur réelle sur $ℝ$ et $υ$ 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.

Reçu le : 2015-02-10
Accepté le : 2016-02-08
Publié le : 2017-07-21

MR

DOI : 10.24033/bsmf.2731

Classification : 34L05, 34B07, 34B20, 37K15
Keywords: Sturm-Liouville problems, quadratic operator pencils, (inverse) spectral theory.
Mots-clés : Problème de Sturm-Liouville, pinceaux quadratiques dopérateurs, théorie spectrale (inverse).

Affiliations des auteurs :

Eckhardt, Jonathan ¹ ; Kostenko, Aleksey ²

¹ School of Computer Science & Informatics, Cardiff University, Queen’s Buildings, 5 The Parade, Roath, Cardiff CF24 3AA, Wales, UK
² Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

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     author = {Eckhardt, Jonathan and Kostenko, Aleksey},
     title = {Quadratic operator pencils associated with the~conservative {Camassa-Holm} flow},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {47--95},
     year = {2017},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {145},
     number = {1},
     doi = {10.24033/bsmf.2731},
     mrnumber = {3636751},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2731/}
}

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Eckhardt, Jonathan; Kostenko, Aleksey. Quadratic operator pencils associated with the conservative Camassa-Holm flow. Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 1, pp. 47-95. doi: 10.24033/bsmf.2731

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