Let X t be a Cauchy process in R , d 1. We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the "Mixed Steklov Problem." Using this we derive a variational characterization for the eigenvalues of the Cauchy process in D. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for Brownian motion. Our results are new even in the simplest geometric setting of the interval (-1, 1) where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions. Such results, although trivial for the Laplacian, take considerable work to prove for the Cauchy processes and remain open for general symmetric #--stable processes. Along the way we present other general properties of the eigenfunctions, such as real analyticity, which even though well known in the case of the Laplacian, are not rarely available for more general symmetric #--stable processes. #

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Abstract. Let D be a domain of finite Lebesgue measure in Rd and let XD t be the symmetric α-stable process killed upon exiting D. Each element of the set {λα i}∞i=1 of eigenvalues associated to XD t, regarded as a function of α ∈ (0, 2), is right continuous. In addition, if D is Lipschitz and bounded, then each λα i is continuous in α and the set of associated eigenfunctions is precompact. We also prove that if D is a domain of finite Lebesgue measure, then for all 0 < α < β ≤ 2 and i ≥ 1, λ α i ≤ λ β] α/β i Previously, this bound had been known only for β = 2 and α rational. 1.

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 [6]. 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.

MÉNDEZ-HERNÁNDEZ Abstract. We prove that the ground state eigenfunction for symmetric stable processes of order α ∈ (0, 2) killed upon leaving the interval (−1, 1) is concave on (−1 1

Spectral gap for stable process on convex planar double symmetric domains

Abstract. It is shown that the second term in the asymptotic expansion as t→0of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order α, for any 0 < α < 2, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.

This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon. 1.

Abstract. We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the half-line (0, ∞) and the interval (−1, 1). This process is related to the square root of one-dimensional Laplacian A = − − d2 dx2 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 ψλ 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, ∞)), and for the distribution of the first exit time from the half-line follow. The formula for ψλ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λn of A in the interval the asymptotic formula λn = nπ π 1 − + O ( ) is 2 8 n derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point. 1.

In this paper we study the behaviour in time of the trace (the partition function) of the heat semigroup associated with symmetric stable processes in domains of Rd. In particular, we show that for domains with the so called R-smoothness property the second terms in the asymptotic as t → 0 involves the surface area of the domain, just as in the case of Brownian motion. Contents §1. Introduction and statement of main result