The Cauchy process and the Steklov problem

Abstract

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.

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

“…Currently, the status of undoubtful relevance have approximate statements (various estimates) pertaining to the asymptotic behavior of eigenfunctions at the well boundaries and estimates, of varied degree of accuracy, of the eigenvalues, c.f. [11] and [12]- [16].…”

“…At this point it is necessary to mention that the Courant-Hilbert nodal line theorem has never been extended to operators which are nonlocal, [12]). As well, no its analog is known in the current context.…”

Fractional calculus for power functions and eigenvalues of the fractional Laplacian

“…are not known explicitly even in the case of d = 1 and B = (−1, 1). A number of methods to study this one-dimensional case, and more general cases, were developed by several authors [1], [2], [3], [4], [5], [7], [13], [14], [24], [25], [26], [27], [28], [29], [32], [36]. The symmetry of eigenfunctions plays an important role in these investigations.…”

$\alpha $-continuity properties of the symmetric $\alpha $-stable process

“…We will need the following lemma; it is formula (2.7) in [5]. Although the authors do not mention the statement concerning continuity in α, it is possible to trace back through the literature they cite to see the statement holds.…”