Spectral properties of the Cauchy process

TL;DR

This paper investigates the spectral properties of the killed one-dimensional Cauchy process on various domains, deriving explicit formulas for eigenfunctions, eigenvalues, and transition densities, and providing numerical bounds for eigenvalues.

Contribution

It provides explicit formulas for eigenfunctions and eigenvalues of the associated operator, and develops numerical methods for precise eigenvalue estimation.

Findings

Explicit formulas for eigenfunctions psi_lambda in the half-line

Asymptotic formula for eigenvalues lambda_n in the interval

Numerical bounds for the first few eigenvalues up to 9th decimal place

Abstract

We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0,infty) and the interval (-1,1). This process is related to the square root of one-dimensional Laplacian A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions psi_lambda of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the half-line (or the heat kernel of A in (0,infty)), and for the distribution of the first exit time from the half-line follow. The formula for psi_lambda is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues lambda_n of A in the interval the asymptotic formula lambda_n = n pi/2 - pi/8 + O(1/n) is derived, and all eigenvalues lambda_n are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues lambda_n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point.