Two methods for solving the Dirac equation without variational collapse

Two special variational techniques, the Lehmann-Maehly (LM) method and the Kato method, recently proposed for solving the one-electron Dirac equation without variational collapse are investigated here in detail. Both methods represent significant progress compared to the traditional variational techniques because each of them provides rigorous upper and lower bounds to relativistic binding energies. A careful theoretical examination, however, reveals that only the LM method can be regarded as a radical solution of all the problems related to variational collapse. A numerical application to the Dirac equation for the hydrogen atom in a uniform magnetic field confirms this conclusion and shows as well that the LM method is also capable of yielding extremely accurate results and that the Kato method, in spite of a few limitations, represents in any case a useful approach.