Upper Bound For The Ratios Of Eigenvalues Of Schrodinger Operators With Nonnegative Single-Barrier Potentials

In this paper we prove the optimal upper bound $\frac{{\lambda }_n}{{\lambda }_m}\le \frac{n^2}{m^2}$ $({\lambda }_n>{\lambda }_m\ge 11\underset{x\in [0,1]}{\sup }q(x))$ for one-dimensional Schrodinger operators with a nonnegative differentiable and single-barrier potential $q(x)$, such that $\mid q^{\mathrm{\prime }}(x)\mid \le q^{\ast },$ where $q^{\ast }=\frac{2}{15}\min \{q(0),q(1)\}$. In particular, if $q(x)$ satisfies the additional condition $\underset{x\in [0,1]}{\sup }q(x)\le \frac{{\pi }^2}{11}$, then \frac{\lambda_{n}}{\lambda_{m}}\leq \frac{n^{2}% }{m^{2}} for $n>m\ge 1.$ For this result, we develop a new approach to study the monotonicity of the modified Pr"{u}fer angle function.