UPPER AND LOWER BOUNDS ON THE ENERGY EIGENVALUES OF THE ONE-ELECTRON DIRAC-HAMILTONIAN

A variational method based on results for self-adjoint operators due to T. Kato [Proc. Phys. Soc. Jpn. 4, 334 (1949)] is developed to calculate upper and lower bounds on the energy eigenvalues for the one-electron Dirac Hamiltonian. The method avoids relativistic variational collapse for any one-electron potential. This result is confirmed analytically in the case of the Coulomb potential and numerically in the case of hydrogenic atoms in very strong magnetic fields for which standard variational techniques cannot preserve bounds. The upper bound thus obtained converges rapidly to the best available numerical results, and provides a very efficient technique for the search of the optimal variational energy by means of a minimization procedure. © 1994 The American Physical Society.