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

Let $M$ be a compact Riemannian manifold with or without boundary, and let $-\mathrm{\Delta }$ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $\lambda \mathrm{\_}i(q)$ the $i$-th eigenvalue of the Schr"{o}dinger type operator $-\mathrm{\Delta }+q$ acting on functions with Dirichlet or Neumann boundary conditions in case $\mathrm{\partial }M\mathrm{\ne }\mathrm{\varnothing }$. We investigate critical potentials of the eigenvalues $\lambda \mathrm{\_}i$ and the eigenvalue gaps $G\mathrm{\_}ij=\lambda \mathrm{\_}j-\lambda \mathrm{\_}i$ 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 $q\in L^{\mathrm{\infty }}(M)$ is critical for the functional $\lambda \mathrm{\_}2$ if and only if, $q$ is smooth, $\lambda \mathrm{\_}2(q)=\lambda \mathrm{\_}3(q)$ and there exist second eigenfunctions $f\mathrm{\_}1,\mathrm{.}\mathrm{.}\mathrm{.},f\mathrm{\_}k$ of $-\mathrm{\Delta }+q$ such that $\mathrm{\Sigma }\mathrm{\_}jf\mathrm{\_}{j}^2=1$. In particular, $\lambda \mathrm{\_}2$ (as well as any $\lambda \mathrm{\_}i$) admits no critical potentials under Dirichlet Boundary conditions. Moreover, the functional $\lambda \mathrm{\_}2$ never admits locally minimizing potentials.