  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.
