Ultrarelativistic bound states in the spherical well

Abstract

We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−∆) 1/2 , whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral data for lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into nonoverlapping, orbitally labelled E (k,l) series. For each orbital label l = 0, 1, 2, ... the label k = 1, 2, ... enumerates consecutive l-th series eigenvalues. Each of them is 2l + 1-degenerate. The l = 0 eigenvalues series E (k,0) are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E (k,0) (d = 3) = E 2k (d = 1). Likewise, the eigenfunctions ψ (k,0) (d = 3) and ψ 2k (d = 1) show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).

Killing (absorption) versus survival in random motion

“…For deceivingly simple problems (albeit in reality, technically quite involved) of the Lvy-stable finite and infinite well or its spherical well analog, numerically accurate and approximate analytic formulas are known for the ground states. We have in hands their shapes (hence the resultant stationary probability densities of the conditioned process) together with corresponding lowest eigenvalues, [38][39][40][41]. Analogous results were established for some Lévy stable oscillators, [37,38,42,43] The solution for the half-line Lévy-stable problem with absorbing barrier (rather involved and available in terms of an approximate analytic expression) is also in existence, [44], and may be used to deduce the process living eternally on the half-line, following our conditioning method.…”

Fractional-Order Operators: Boundary Problems, Heat Equations

“…e.g. [AJS18,DKK17,DG17,ZG16]). Each of the methods allow different types of generalizations of the fractional Laplacian, and solve a variety of problems, primarily in low-regularity spaces.…”

Ultrarelativistic bound states in the shallow spherical well

“…The ultrarelativistic operator is nonlocal (quasirelativistic and α-stable operators likewise) and we employ its integral definition (involving a suitable function f (x), with x ∈ R d ), that is valid in space dimensions d ≥ 1, [9,10]:…”

“…Eq. ( 1) derives from the more general integral definition of the α ∈ (0, 2) -Lévy stable rotationally symetric operator [9] and is hereby specialized to α = 1. The (Lévy measure) normalization coefficient…”

“…Our departure point is the recent paper [9], where the employed exterior Dirichlet boundary data were interpreted as the infinite spherical well enclosure for the 3D ultrarelativistic operator. Spectral links with the 1D infinite well problem (that actually has been solved in [11], see also [12,13] and [14]) have proved to be instrumental for the 3D derivations in Ref.…”