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 $-Δ$ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $λ\_i(q)$ the $i$-th eigenvalue of the Schrödinger type operator $-Δ+ q$ acting on functions with Dirichlet or Neumann boundary conditions in case $\partial M \neq \emptyset$. We investigate critical potentials of the eigenvalues $λ\_i$ and the eigenvalue gaps $G\_{ij}=λ\_j -λ\_i$ 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 $λ\_2$ if and only if, $q$ is smooth, $λ\_2(q)=λ\_3(q)$ and there exist second eigenfunctions $f\_1 ,...,f\_k$ of $-Δ+ q$ such that $Σ\_j f\_j^2 = 1$. In particular, $λ\_2$ (as well as any $λ\_i$) admits no critical potentials under Dirichlet Boundary conditions. Moreover, the functional $λ\_2$ never admits locally minimizing potentials.