Optimal potentials for Schrödinger operators

Giuseppe Buttazzo; Augusto Gerolin; Berardo Ruffini; Bozhidar Velichkov

doi:10.5802/jep.4

Optimal potentials for Schrödinger operators

Giuseppe Buttazzo ¹; Augusto Gerolin ¹; Berardo Ruffini ²; Bozhidar Velichkov ¹

¹ Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo 5, 56127 Pisa, Italy
² Laboratoire Jean Kuntzmann, Université de Grenoble BP 53, 38041 Grenoble Cedex 9, France

Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 71-100.

Abstract
Résumé

We consider the Schrödinger operator $- Δ + V (x)$ on $H_{0}^{1} (Ω)$ , where $Ω$ is a given domain of $ℝ^{d}$ . Our goal is to study some optimization problems where an optimal potential $V \geq 0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Nous considérons l’opérateur de Schrödinger $- Δ + V (x)$ sur $H_{0}^{1} (Ω)$ , où $Ω$ est un domaine fixé de $ℝ^{d}$ . Nous étudions certains problèmes d’optimisation pour lesquels un potentiel optimal $V \geq 0$ doit être déterminé dans une certaine classe admissible et pour certains critères d’optimisation tels que l’énergie ou les valeurs propres de Dirichlet.

Received: 2013-11-28
Accepted: 2014-05-01
Published online: 2014-08-26

Zbl

DOI: 10.5802/jep.4

Classification: 49J45, 35J10, 49R05, 35P15, 35J05
Keywords: Schrödinger operators, optimal potentials, spectral optimization, capacity
Mot clés : Opérateurs de Schrödinger, potentiels optimaux, optimisation spectrale, capacité

Author's affiliations:

Giuseppe Buttazzo ¹; Augusto Gerolin ¹; Berardo Ruffini ²; Bozhidar Velichkov ¹

¹ Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo 5, 56127 Pisa, Italy
² Laboratoire Jean Kuntzmann, Université de Grenoble BP 53, 38041 Grenoble Cedex 9, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Giuseppe Buttazzo and Augusto Gerolin and Berardo Ruffini and Bozhidar Velichkov},
     title = {Optimal potentials for {Schr\"odinger~operators}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {71--100},
     publisher = {\'Ecole polytechnique},
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Giuseppe Buttazzo; Augusto Gerolin; Berardo Ruffini; Bozhidar Velichkov. Optimal potentials for Schrödinger operators. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 71-100. doi : 10.5802/jep.4. https://jep.centre-mersenne.org/articles/10.5802/jep.4/

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