Lower Bound For The Ratios Of Eigenvalues Of Schr\"odinger Equations With Nonpositive Single-Barrier Potentials

TL;DR

This paper establishes a lower bound for ratios of eigenvalues in Schrödinger equations with nonpositive, single-barrier potentials, extending previous bounds known for nonnegative, single-well potentials, using a new approach involving the Pr"ufer angle function.

Contribution

It proves a lower bound for eigenvalue ratios under nonpositive, single-barrier potentials, complementing existing upper bounds for nonnegative, single-well potentials, with a novel method involving the Pr"ufer angle.

Findings

Eigenvalue ratios are bounded below by n^2/m^2 for certain potentials.

The result applies when the potential's minimum at the endpoints is bounded by π^2/3.

A new approach using the monotonicity of the modified Pr"ufer angle is developed.

Abstract

Horv\'ath and Kiss [Proc. Amer. Math. Soc., 2005] proved the upper bound estimate $\frac{\lambda _{n}}{\lambda _{m}}\leq \frac{n^{2}}{m^{2}}$ $ (n>m\geq 1) $ for Dirichlet eigenvalue ratios of the Schr\"odinger problem $-y''+q(x)y=\lambda y$ 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 $\frac{\lambda_{n}}{\lambda_{m}}\geq \frac{n^{2}}{m^{2}}$ for $\lambda_n>\lambda_m \geq -2q^*$, where $q^{\ast}=\min\{q(0), q(1)\}$. In particular, if $q(x)$ satisfies the additional condition $\mid q^{\ast} \mid\leq \frac{\pi^{2}}{3}$, then $\lambda _{1}>0$ and $\frac{\lambda _{n}}{\lambda _{m}}\geq \frac{n^{2}%}{m^{2}}$ for $n>m\geq 1.$ For this result, we develop a new approach to study the monotonicity of the modified Pr\"ufer angle function.