Eigenvalue gaps for the Cauchy process and a Poincaré inequality

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 [R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211 (2004) 355–423]. From this, a variational characterization for the eigenvalues λn, n⩾1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between λn and λ1. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for λ∗−λ1 where λ∗ is the eigenvalue corresponding to the “first” antisymmetric eigenfunction for D. The proof is based on a variational characterization of λ∗−λ1 and on a weighted Poincaré-type inequality. The Poincaré inequality is valid for all α symmetric stable processes, 0<α⩽2, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap λ2−λ1 in bounded convex domains.