Partial differential equations/Differential geometry
Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders
Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 813-818.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.06.018
Dolbeault, Jean 1; Esteban, Maria J. 1; Loss, Michael 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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     title = {Keller{\textendash}Lieb{\textendash}Thirring inequalities for {Schr\"odinger} operators on cylinders},
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Dolbeault, Jean; Esteban, Maria J.; Loss, Michael. 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. http://www.numdam.org/articles/10.1016/j.crma.2015.06.018/

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