A second eigenvalue bound for the Dirichlet Schroedinger operator

Abstract

Let ${\lambda }_i(\Omega ,V)$ be the $i$th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset {\mathbb{R}}^n$ and with the positive potential $V$. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_{\star }$, we prove that ${\lambda }_2(\Omega ,V)\le {\lambda }_2(S_1,V_{\star })$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition ${\lambda }_1(\Omega ,V)={\lambda }_1(S_1,V_{\star })$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that ${\lambda }_2(B_R,V)/{\lambda }_1(B_R,V)$ decreases when the radius $R$ of the ball $B_R$ increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.