We consider the problem of minimizing the eigenvalues of the Schr\"{ o}dinger operator $H=-\Delta+\alpha F(\ka)$ ($\alpha>0$) on a compact $n-$manifold subject to the restriction that $\ka$ has a given fixed average $\ka_{ 0}$. In the one-dimensional case our results imply in particular that for $F(\ka)=\ka^{ 2}$ the constant potential fails to minimize the principal eigenvalue for $\alpha>\alpha_{ c}=\mu_{ 1}/(4\ka_{ 0}^{ 2})$, where $\mu_{ 1}$ is the first nonzero eigenvalue of $-\Delta$. This complements a result by Exner, Harrell and Loss (math-ph/9901022), showing that the critical value where the circle stops being a minimizer for a class of Schr\"{ o}dinger operators penalized by curvature is given by $\alpha_{ c}$. Furthermore, we show that the value of $\mu_{ 1}/4$ remains the infimum for all $\alpha>\alpha_{ c}$. Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential. In higher dimensions we prove a (weak) local version of these results for a general class of potentials $F(\ka)$, and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding eigenvalues of the Laplace-Beltrami operator and is never attained.
Categories: math-ph math.MP math.SP