Eigenvalue ratios for Schrödinger operators with indefinite potentials

We consider the Schrödinger equation −y′′+q(x)y=λy, on a finite interval with Dirichlet boundary conditions, where q(x) is of indefinite sign. In the case of symmetric potentials, we prove the optimal lower bound λnλ1≥n2 (resp. upper bound λnλ1≤n2) for single-well q with λ1>0 and μ1≤0 (resp. single-barrier q and μ1≥0), where μ1 is the first eigenvalue of the Neumann boundary problem. In the case of nonsymmetric potentials, we prove the optimal lower bound λ2λ1≥4 for single-well q with transition point at x=12, λ1>0 and μ=max(μˆ1,μ̃1)≤0, where μˆ1 and μ̃1 are the first eigenvalues of the Neumann boundary problems defined on [0,12] and [12,1], respectively.