Rafael D. Benguria; Helmut Linde
A second eigenvalue bound for the Dirichlet Schroedinger operator
(261K, Postscript)

ABSTRACT.  Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a 
bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the 
Payne-P\'olya-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.