Relativistic Comparison Theorems

Comparison theorems are established for the Dirac and Klein–Gordon equations. We suppose that V^{(1)}® and V^{(2)}® 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)}® \leq V^{(2)}®, 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.