  A connection between the semigroup of the Cauchy process killed upon exiting
a domain $D$ and a mixed boundary value problem for the Laplacian in one
dimension higher known as the "mixed Steklov problem," was established in a
previous paper of the authors. From this, a variational characterization for
the eigenvalues $\lambda_n$, $n\geq 1$, of the Cauchy process in $D$ was
obtained. In this paper we obtain a variational characterization of the
difference between $\lambda_n$ and $\lambda_1$. We study bounded convex domains
which are symmetric with respect to one of the coordinate axis and obtain lower
bound estimates for $\lambda_* - \lambda_1$ where $\lambda_*$ is the eigenvalue
corresponding to the "first" antisymmetric eigenfunction for $D$. The proof is
based on a variational characterization of $\lambda_* - \lambda_1$ and on a
weighted Poincar\'e--type inequality. The Poincar\'e inequality is valid for
all $\alpha$ symmetric stable processes, $0<\alpha\leq 2$, and any other
process obtained from Brownian motion by subordination. We also prove upper
bound estimates for the spectral gap $\lambda_2-\lambda_1$ in bounded convex
domains.
