On minimal eigenvalues of Schrodinger operators on manifolds

TL;DR

This paper investigates the minimization of eigenvalues of Schrödinger operators on manifolds, revealing conditions under which constant potentials are optimal and providing bounds for eigenvalues in various dimensions.

Contribution

It establishes new criteria for when constant potentials minimize eigenvalues, extends results to higher dimensions, and offers sharp bounds for Schrödinger operators on manifolds.

Findings

Constant potential fails to minimize principal eigenvalue beyond a critical parameter.

The critical value for minimization is explicitly characterized.

A sharp lower bound for the principal eigenvalue is derived.

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.