Relativistic comparison theorems
Comparison theorems are established for the Dirac and Klein-Gordon equations. We suppose that V{sup (1)}(r) and V{sup (2)}(r) are two real attractive central potentials in d dimensions that support discrete Dirac eigenvalues E{sub k{sub d{nu}}{sup (1)}} and E{sub k{sub d{nu}}{sup (2)}}. We prove that if V{sup (1)}(r){<=}V{sup (2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered E{sub k{sub d{nu}}{sup (1)}}{<=}E{sub k{sub d{nu}}{sup (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.