Critical potentials of the eigenvalues and eigenvalue gaps of Schr\"{o}dinger operators

Abstract

Let $M$ be a compact Riemannian manifold with or without boundary, and let $-\Delta $ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $\lambda\_i(q)$ the $i$-th eigenvalue of the Schr\"{o}dinger type operator $-\Delta + q$ acting on functions with Dirichlet or Neumann boundary conditions in case $\partial M \neq \emptyset$. We investigate critical potentials of the eigenvalues $\lambda\_i$ and the eigenvalue gaps $G\_{ij}=\lambda\_j -\lambda\_i$ considered as functionals on the set of bounded potentials having a given mean value on $M$. We give necessary and sufficient conditions for a potential $q$ to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential $q \in L^\infty (M)$ is critical for the functional $\lambda\_2$ if and only if, $q$ is smooth, $\lambda\_2(q)=\lambda\_3(q)$ and there exist second eigenfunctions $f\_1 ,...,f\_k$ of $-\Delta + q$ such that $\Sigma\_j f\_j^2 = 1$. In particular, $\lambda\_2$ (as well as any $\lambda\_i$) admits no critical potentials under Dirichlet Boundary conditions. Moreover, the functional $\lambda\_2$ never admits locally minimizing potentials.