Shifted large-Nexpansion for the energy levels of relativistic particles

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Abstract
The shifted large-N expansion method, which was developed to obtain accurate energy eigenvalues for nonrelativistic-potential problems, has been extended to deal with a relativistic particle (with or without spin) bound in a spherically symmetric potential. The calculations are carried out for any arbitrary quantum state using expansion in terms of a parameter 1/k¯, where k¯ contains the dimension of the space N and the so-called shift parameter. Similar to the work of T. Imbo, A. Pagnamenta, and U. Sukhatme [Phys. Rev. D 29, 1669 (1984)] we suggest determination of the shift parameter in such a way that the exact analytic result for the nonrelativistic Coulomb binding energy is restored. As a consequence of this choice, we obtain also a highly convergent expansion for the relativistic part of the energy eigenvalue. Although the formalism is developed for spin-zero and spin-(1/2 particles in any arbitrary spherically symmetric potential, it is illustrated for the Coulomb potential as a special case. Our results are consistently better than those previously obtained by using the unshifted 1/N expansion technique. The shifted 1/N expansion is seen to be applicable to a much wider class of relativistic potentials which may have applications in atomic processes. A few interesting aspects of our approach are briefly discussed.