Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders

Jean Dolbeault; Maria J. Esteban; Michael Loss

doi:10.1016/j.crma.2015.06.018

Partial differential equations/Differential geometry

Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders
[Inégalités de Keller–Lieb–Thirring pour des opérateurs de Schrödinger sur des cylindres]

Présenté par : Haïm Brézis

Jean Dolbeault ¹ ; Maria J. Esteban ¹ ; Michael Loss ²

¹ Ceremade UMR CNRS No. 7534, Université Paris-Dauphine, place de Lattre-de-Tassigny, 75775 Paris cedex 16, France
² Skiles Building, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 813-818.

Résumé
Abstract

Cette note est consacrée à des estimations spectrales de Keller–Lieb–Thirring pour des opérateurs de Schrödinger sur des cylindres infinis : la valeur absolue de l'état fondamental est bornée par une fonction d'une norme du potentiel. Il est montré que les potentiels optimaux de petite norme ne dépendent que d'une seule variable : il s'agit d'un résultat de symétrie. La preuve provient d'un argument de perturbation qui repose sur des résultats de rigidité récents pour des équations elliptiques non linéaires sur des cylindres. À l'inverse, les potentiels optimaux de grande norme qui ne dépendent que d'une seule variable sont instables : cela fournit un résultat de brisure de symétrie. La valeur optimale qui sépare les deux régimes est établie dans le cas du produit d'une sphère et d'une droite.

This note is devoted to Keller–Lieb–Thirring spectral estimates for Schrödinger operators on infinite cylinders: the absolute value of the ground state level is bounded by a function of a norm of the potential. Optimal potentials with small norms are shown to depend on a single variable: this is a symmetry result. The proof is a perturbation argument based on recent rigidity results for nonlinear elliptic equations on cylinders. Conversely, optimal single variable potentials with large norms must be unstable: this provides a symmetry breaking result. The optimal threshold between the two regimes is established in the case of the product of a sphere by a line.

Reçu le : 2015-03-30
Accepté le : 2015-06-12
Publié le : 2015-07-26

DOI : 10.1016/j.crma.2015.06.018

Affiliations des auteurs :

Jean Dolbeault ¹ ; Maria J. Esteban ¹ ; Michael Loss ²

¹ Ceremade UMR CNRS No. 7534, Université Paris-Dauphine, place de Lattre-de-Tassigny, 75775 Paris cedex 16, France
² Skiles Building, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

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     author = {Jean Dolbeault and Maria J. Esteban and Michael Loss},
     title = {Keller{\textendash}Lieb{\textendash}Thirring inequalities for {Schr\"odinger} operators on cylinders},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {813--818},
     publisher = {Elsevier},
     volume = {353},
     number = {9},
     year = {2015},
     doi = {10.1016/j.crma.2015.06.018},
     language = {en},
}

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Jean Dolbeault; Maria J. Esteban; Michael Loss. Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders. Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 813-818. doi : 10.1016/j.crma.2015.06.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.06.018/

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