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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.
Dolbeault, Jean 1 ; Esteban, Maria J. 1 ; Loss, Michael 2
@article{CRMATH_2015__353_9_813_0,
author = {Dolbeault, Jean and Esteban, Maria J. and Loss, Michael},
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},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2015.06.018/}
}
TY - JOUR AU - Dolbeault, Jean AU - Esteban, Maria J. AU - Loss, Michael TI - Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders JO - Comptes Rendus. Mathématique PY - 2015 SP - 813 EP - 818 VL - 353 IS - 9 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2015.06.018/ DO - 10.1016/j.crma.2015.06.018 LA - en ID - CRMATH_2015__353_9_813_0 ER -
%0 Journal Article %A Dolbeault, Jean %A Esteban, Maria J. %A Loss, Michael %T Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders %J Comptes Rendus. Mathématique %D 2015 %P 813-818 %V 353 %N 9 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2015.06.018/ %R 10.1016/j.crma.2015.06.018 %G en %F CRMATH_2015__353_9_813_0
Dolbeault, Jean; Esteban, Maria J.; Loss, Michael. Keller–Lieb–Thirring inequalities for Schrödinger operators on cylinders. Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 813-818. doi : 10.1016/j.crma.2015.06.018. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2015.06.018/
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