Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

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 λi(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 λi and the eigenvalue gaps Gij = λj − λi 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 λ2 if and only if q is smooth, λ2(q) = λ3(q) and there exist second eigenfunctions f1, . . . , fk of −Δ+q such that j f 2 j = 1. In particular, λ2 (as well as any λi ) admits no critical potentials under Dirichlet boundary conditions.Moreover, the functional λ2 never admits locally minimizing potentials.  2005 Elsevier Inc. All rights reserved.