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Title: Ultrarelativistic bound states in the spherical well

Abstract

We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−Δ){sup 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 non-overlapping, orbitally labelled E{sub (k,l)} series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive lth series eigenvalues. Each of them is 2l + 1-degenerate. The l = 0 eigenvalues series E{sub (k,0)} are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E{sub (k,0)}(d = 3) = E{sub 2k}(d = 1). Likewise, the eigenfunctions ψ{sub (k,0)}(d = 3) and ψ{sub 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 themore » universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).« less

Authors:
;  [1]
  1. Institute of Physics, University of Opole, 45-052 Opole (Poland)
Publication Date:
OSTI Identifier:
22596630
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 57; Journal Issue: 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; EIGENFUNCTIONS; EIGENVALUES; RELATIVISTIC RANGE; SPHERICAL CONFIGURATION

Citation Formats

Żaba, Mariusz, and Garbaczewski, Piotr. Ultrarelativistic bound states in the spherical well. United States: N. p., 2016. Web. doi:10.1063/1.4955168.
Żaba, Mariusz, & Garbaczewski, Piotr. Ultrarelativistic bound states in the spherical well. United States. doi:10.1063/1.4955168.
Żaba, Mariusz, and Garbaczewski, Piotr. Fri . "Ultrarelativistic bound states in the spherical well". United States. doi:10.1063/1.4955168.
@article{osti_22596630,
title = {Ultrarelativistic bound states in the spherical well},
author = {Żaba, Mariusz and Garbaczewski, Piotr},
abstractNote = {We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−Δ){sup 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 non-overlapping, orbitally labelled E{sub (k,l)} series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive lth series eigenvalues. Each of them is 2l + 1-degenerate. The l = 0 eigenvalues series E{sub (k,0)} are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E{sub (k,0)}(d = 3) = E{sub 2k}(d = 1). Likewise, the eigenfunctions ψ{sub (k,0)}(d = 3) and ψ{sub 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).},
doi = {10.1063/1.4955168},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 7,
volume = 57,
place = {United States},
year = {2016},
month = {7}
}