Eigenvalue ratios for Sturm–Liouville problems

Abstract

In this paper, we prove the optimal lower bound λ2λ1≥4 of the Sturm-Liouville problem −(p(x)y′)′ + q(x)y = λρ(x)y, with Dirichlet boundary conditions for single-well potential q and single-barrier function pρ on [0,1]. In the case of symmetric coefficients, we establish the lower bound λnλ1≥n2 for single-well q and single-barrier function pρ with λ1 > 0 and μ1 ≤ 0, where μ1 is the first eigenvalue of the Neumann boundary problem. All the above estimates are given for q ≤ 0 without assumptions on the monotonicity of q.