Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Details Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

2005-06-10

arxiv.org/abs/math/0506195v1

Journal of Mathematical Analysis and Applications 314 (2006) 195-209

Mathematics

Metric Geometry

Scientific paper

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.

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