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

摘要

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 Schr\"{o}dinger operators with a nonnegativedifferentiable and single-barrier potential $q(x)$, such that $\mid q'(x)\mid\leq q^{*},$ where $q^{*}=\frac{2}{15}\inf\{q(0) ,q(1)\}$. 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 newapproach to study the monotonicity of the modified Pr\"{u}fer angle function.