Spectral optimization problems for Schrödinger operators

In this chapter we consider Schrödinger operators of the form −∆+V(x) on the Sobolev space H_0^1(D), where D is an open subset of R^d. We are interested in finding optimal potentials for some suitable criteria; the optimization problems we deal with are then written as min {F(V) : V∈V} where F is a suitable cost functional and V is a suitable class of admissible potentials. For simplicity, we consider the case when D is bounded and V ≥ 0; under these conditions the resolvent operator of −∆ + V(x) is compact and the spectrum λ(V) of the Schrödinger operator is discrete and consists of an increasing sequence of positive eigenvalues λ(V) = (λ_1(V), λ_2(V), ...).