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.
Accepted:
Published online:
Jean Dolbeault 1; Maria J. Esteban 1; Michael Loss 2
@article{CRMATH_2015__353_9_813_0, 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}, }
TY - JOUR AU - Jean Dolbeault AU - Maria J. Esteban AU - Michael Loss 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 DO - 10.1016/j.crma.2015.06.018 LA - en ID - CRMATH_2015__353_9_813_0 ER -
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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