Eigenvalue gaps for the Cauchy process and a Poincar\'e inequality

TL;DR

This paper investigates eigenvalue gaps for the Cauchy process in convex domains, establishing variational characterizations, lower bounds for antisymmetric eigenvalues, and upper bounds for spectral gaps, with implications for Poincaré inequalities.

Contribution

It introduces a variational characterization of eigenvalue differences for the Cauchy process and derives bounds for spectral gaps in convex domains, extending Poincaré inequalities to stable processes.

Findings

Lower bounds for eigenvalue differences in symmetric convex domains.

Upper bounds for the spectral gap in bounded convex domains.

A Poincaré inequality valid for all symmetric stable processes.

Abstract

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.