A second eigenvalue bound for the Dirichlet Schroedinger operator

Abstract

Let $λ_i(Ω,V)$ be the $i$th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $Ω\subset \R^n$ 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 $λ_2(Ω,V) \le λ_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $λ_1(Ω,V) = λ_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $λ_2(B_R, V) / λ_1(B_R, V)$ 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.