On minimal eigenvalues of Schrodinger operators on manifolds

Abstract

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.Comment: 7 page