On minimal eigenvalues of Schrodinger operators on manifolds

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

Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in R^d

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues for a large class of domains, under very mild assumptions. As an application, we derive a three--term asymptotic expansion for the first eigenvalue of d-dimensional ellipsoids