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Let X t be a Cauchy process in R d ; dX1: 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 ðÀ1; 1Þ 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 a-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 available for more general symmetric a-stable processes. r 2004 Elsevier Inc. All rights reserved.

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c ∪{∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.