Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator

Abstract:

In this paper the authors prove the following theorem: Let $\Omega$ be a smooth strictly convex bounded domain in ${R^n}$ and $V:\Omega \to R$ a nonnegative convex function. Suppose ${\lambda _1}$ and ${\lambda _2}$ are the first and second nonzero eigenvalues of the equation \[ - \Delta f + Vf = \lambda f,\qquad f{|_{\partial \Omega }} \equiv 0.\] Then ${\lambda _2} - {\lambda _1} \geqslant {\pi ^2}/{d^2}$, where $d$ is the diameter of $\Omega$.