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 ith eigenvalue of the Schrödinger type operator − Δ + q acting on functions with Dirichlet or Neumann boundary conditions in case ∂ M ≠ ∅ . We investigate critical potentials of the eigenvalues λ i and the eigenvalue gaps G i j = 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 ∈ L ∞ ( 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 ∑ 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.