Optimal potentials for Schrödinger operators

Abstract

We consider the Schrödinger operator −∆ + V (x) on H 1 0 (Ω), where Ω is a given domain of R d . Our goal is to study some optimization problems where an optimal potential V ≥ 0 has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Existence and Regularity Results for Some Shape Optimization Problems

“…Optimization problems for Schrödinger operators (for k = 1, 2 the result was proved in [34], while for generic k ∈ N the existence is proved in [26]) min λ k (− +V ) : V : R d →[0, +∞] measurable, Optimization problems for Schrödinger operators (for k = 1, 2 the result was proved in [34], while for generic k ∈ N the existence is proved in [26]) min λ k (− +V ) : V : R d →[0, +∞] measurable,…”

“…We present the recent results from [20]- [81] [25,59] and [34]- [26], introducing the existence and regularity techniques involving the results from the previous chapters and simplifying some of the original proofs. We present the recent results from [20]- [81] [25,59] and [34]- [26], introducing the existence and regularity techniques involving the results from the previous chapters and simplifying some of the original proofs.…”

“…In fact, let v = w 24 = w 34 : [0, l 24 +l34 2 ] → R be an increasing function such that 2|{v ≥ τ }| = |{w 24 ≥ τ }| + |{w 34 ≥ τ }|. Without loss of generality, we can suppose that the maximum of the energy function w on is achieved on the edge e 14 .…”

Spectral Optimization Problems for Potentials and Measures

“…As in the case of shapes, when the competing potentials are assumed to be supported in a given bounded set D ⊂ R d , a general result (see Theorem 4.1 of [13]) provides the existence of an optimal potential under the assumptions:…”

“…The main purpose of the paper is to consider optimization problems for Schrödinger potentials min F (V ) : 2) in the spirit of [13] (see the definition of the class V below). As in the case of shapes, when the competing potentials are assumed to be supported in a given bounded set D ⊂ R d , a general result (see Theorem 4.1 of [13]) provides the existence of an optimal potential under the assumptions:…”