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

Optimal design problems for Schr\"odinger operators with noncompact resolvents