Lower Bound For The Ratios Of Eigenvalues Of Schrödinger Equations With Nonpositive Single-Barrier Potentials

Abstract

Horváth and Kiss [Proc. Amer. Math. Soc., 2005] proved the upper bound estimate $\frac{{\lambda }_n}{{\lambda }_m}\le \frac{n^2}{m^2}$ $(n>m\ge 1)$ for Dirichlet eigenvalue ratios of the Schrödinger problem $-y^{\prime \prime }+q(x)y=\lambda y$ 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 $\frac{{\lambda }_n}{{\lambda }_m}\ge \frac{n^2}{m^2}$ for ${\lambda }_n>{\lambda }_m\ge -2q^{\ast }$, where $q^{\ast }=\min \{q(0),q(1)\}$. In particular, if $q(x)$ satisfies the additional condition $\mid q^{\ast }\mid \le \frac{{\pi }^2}{3}$, then ${\lambda }_1>0$ and $\frac{λ_{n}}{λ_{m}}\geq \frac{n^{2}%}{m^{2}}$ for $n>m\ge 1.$ For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.

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