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

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