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.
Categories: math-ph math.MP physics.atom-ph quant-ph