  In the present paper we consider spectral optimization problems involving the
Schr\"odinger operator $-\Delta +\mu$ on $\R^d$, the prototype being the
minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ 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 $\ds \int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of
global solutions in $\R^d$ and that the optimal potentials or measures are
equal to $+\infty$ outside a compact set.
