Optimal potentials for Schrödinger operators

Buttazzo, Giuseppe; Gerolin, Augusto; Ruffini, Berardo; Velichkov, Bozhidar

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

Nous considérons l’opérateur de Schrödinger $-\Delta +V\left(x\right)$ sur $H_0^1\left(\Omega \right)$, où $\Omega$ est un domaine fixé de ${ℝ}^d$. Nous étudions certains problèmes d’optimisation pour lesquels un potentiel optimal $V\ge 0$ doit être déterminé dans une certaine classe admissible et pour certains critères d’optimisation tels que l’énergie ou les valeurs propres de Dirichlet.

We consider the Schrödinger operator $-\Delta +V\left(x\right)$ on $H_0^1\left(\Omega \right)$, where $\Omega$ is a given domain of ${ℝ}^d$. Our goal is to study some optimization problems where an optimal potential $V\ge 0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

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

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