Optimal design problems for Schrodinger operators with noncompact resolvents

We consider optimization problems for cost functionals which depend on the negative spectrum of Schrodinger operators of the form $-Delta+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under suitable assumptions the existence of an optimal potential is shown. This can be applied to interesting cases such as costs functions involving finitely many negative eigenvalues.