Lower bound for the ratios of eigenvalues of Schrödinger with nonpositive single‐barrier potentials

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Horváth and Kiss (Proc. Amer. Math. Soc., 2005) proved the upper bound estimate for Dirichlet eigenvalue ratios of the Schrödinger problem −y′′ + q(x)y = λy with nonnegative and single‐well potential q. In this paper, we prove that if q(x) is a nonpositive, continuous, and single‐barrier potential, then for λn > λm≥ − 2q∗, where . In particular, if q(x) satisfies the additional condition , then λ1 > 0 and for n > m ≥ 1. For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.