Spectral properties of the Cauchy process on half-line and interval

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 = − − 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 ψ λ 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 ψ λ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λ n of A in the interval the asymptotic formula λ n = nπ 2 − π 8 + O( 1 n ) is derived, and all eigenvalues λ n are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λ n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point.

Fractional calculus for power functions and eigenvalues of the fractional Laplacian

“…It is conjectured, but not yet proved in the full range of α ∈ (0, 2) and d, that λ * = λ 2 . In the classical case (α = 2), and also in the 1-dimensional case for α ≥ 1 [26], we do have λ * = λ 2 . It is natural to ask whether λ * = λ 2 .…”

“…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.…”

Ultrarelativistic bound states in the spherical well

“…The essential progress has been made just recently, [11][12][13]. The infinite fractional well in d = 1 has been studied primarily by mathematicians and preliminary attempts were made to attack the fully-fledged d = 3 spectral problem, [14]- [20].…”

Radiative Transfer in Half Spaces of Arbitrary Dimension

“…We now show that Flatland H can be expressed analytically for all 0 < µ < 1, 0 < c < 1 by exploiting an integral form presented by Fock [18] for a diffraction problem with the MacDonald kernel of Eq.(23). Fock found (see also [65]), Similar expressions involving Li 2 (z) and 3 F 2 also arise in the case of a Cauchy random flight in a 1D rod [95] and path integrals over half spaces [96]. These analytic forms have advantages relative to the numerical evaluation of integrals using quadrature and related methods.…”