Spectral Optimization Problems for Potentials and Measures

In the present paper we consider spectral optimization problems involving the Schrödinger operator −∆ + μ on R, the prototype being the minimization of the k the eigenvalue λk(μ). Here μ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential V which satisfies the integral constraint ∫ V −pdx ≤ m with 0 < p < 1. We prove the existence of global solutions in R and that the optimal potentials or measures are equal to +∞ outside a compact set.