Relativistic comparison theorems

Richard L. Hall
Phys. Rev. A 81, 052101 – Published 5 May 2010

Abstract

Comparison theorems are established for the Dirac and Klein-Gordon equations. We suppose that V(1)(r) and V(2)(r) are two real attractive central potentials in d dimensions that support discrete Dirac eigenvalues Ekdν(1) and Ekdν(2). We prove that if V(1)(r)V(2)(r), then each of the corresponding discrete eigenvalue pairs is ordered Ekdν(1)Ekdν(2). This result generalizes an earlier, more restrictive theorem that required the wave functions to be node-free. For the the Klein-Gordon equation, similar reasoning also leads to a comparison theorem provided in this case that the potentials are negative and the eigenvalues are positive.

  • Received 5 February 2010

DOI:https://doi.org/10.1103/PhysRevA.81.052101

©2010 American Physical Society

Authors & Affiliations

Richard L. Hall*

  • Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montreal, Quebec H3G 1M8, Canada

  • *rhall@mathstat.concordia.ca

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Vol. 81, Iss. 5 — May 2010

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