Critical potentials of the eigenvalue ratios of Schrödinger operators

In this paper, we consider an operator HH defined on a compact smooth nn-manifold MM by H=−Δ+q acting on functions with Dirichlet or Neumann boundary conditions in the case ∂M≠0̸∂M≠0̸ and with eigenvalues λ1(q)<λ2(q)≤λ3(q)≤⋯λ1(q)<λ2(q)≤λ3(q)≤⋯. We investigate critical potentials of the ratio of two consecutive eigenvalues considered as functionals on the set of bounded potentials having a given mean value on MM. We obtain necessary and sufficient conditions for a potential to be a critical point of such a functional.