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

## TL;DR

This paper establishes an optimal upper bound for the ratio of eigenvalues of one-dimensional Schrödinger operators with certain nonnegative single-barrier potentials, using a novel approach to analyze the Pr"ufer angle function.

## Contribution

It introduces a new method to prove the eigenvalue ratio bound for Schrödinger operators with specific potential conditions, improving understanding of spectral properties.

## Key findings

- Proves the bound $rac{
abla_{n}}{
abla_{m}} extless rac{n^{2}}{m^{2}}$ under given conditions.
- Establishes the bound holds for potentials with $ ext{sup}_{x ext{ in }[0,1]} q(x) extless rac{ ext{ extpi}^{2}}{11}$.
- Develops a new approach to analyze the monotonicity of the modified Prüfer angle function.

## Abstract

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.