Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger 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 Schrödinger type operator $-\Delta +q$ acting on functions with Dirichlet or Neumann boundary conditions in case $\partial M\ne \varnothing$. 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 $\Sigma \_jf\_j^2=1$. 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.