Relativistic Comparison Theorems

TL;DR

This paper establishes comparison theorems for Dirac and Klein-Gordon equations, showing how eigenvalues relate when potentials are ordered, extending previous results to cases with wave functions that have nodes.

Contribution

It generalizes existing comparison theorems for relativistic equations by removing the restriction on wave function nodes and applies to both Dirac and Klein-Gordon equations.

Findings

Eigenvalues are ordered when potentials are ordered, E^{(1)}_{k_d u} q; E^{(2)}_{k_d u} for Dirac.

Similar comparison theorem holds for Klein-Gordon with negative potentials and positive eigenvalues.

Extension of previous theorems to include wave functions with nodes.

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 E^{(1)}_{k_d\nu} and E^{(2)}_{k_d\nu}. We prove that if V^{(1)}(r) \leq V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered E^{(1)}_{k_d\nu} \leq E^{(2)}_{k_d\nu}. 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.