On minimal eigenvalues of Schrodinger operators on manifolds

Abstract

We consider the problem of minimizing the eigenvalues of the Schrödinger operator $H=-Δ+αF(\ka)$ ($α>0$) 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 $α>α_{c}=μ_{1}/(4\ka_{0}^{2})$, where $μ_{1}$ is the first nonzero eigenvalue of $-Δ$. 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 Schrödinger operators penalized by curvature is given by $α_{c}$. Furthermore, we show that the value of $μ_{1}/4$ remains the infimum for all $α>α_{c}$. 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.