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The difficulties associated with application of the shifted large-N technique to the Dirac equation have been resolved by applying the method to the Klein-Gordon equation in which a spin-orbit interaction term is included analogous to Pauli theory. Explicit analytical expressions for the relativistic screened Coulomb bound-state energies, radial wave functions, and normalizations are given.For the point-Coulomb problem, we restore exact results for the relativistic binding energies and almost exact wave functions. The 1/N expansion results are then compared with the exact numerical solutions as well as with those obtained in other analytical methods for a number of screened Coulomb potentials and for a wide range of atomic numbers Z. In general, excellent agreement is found. In contrast to the limited applicability of the usual perturbative methods, our technique is found to be flexible and may be extended to a more general class of relativistic potentials that has applications in atomic and quarkonium physics. Encouraging aspects of the present approach are also briefly discussed.

The shifted large-N expansion method, which was developed to obtain accurate energy eigenvalues for nonrelativistic-potential problems, has been extended to deal with 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] 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-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.

a b s t r a c tThis paper deals with derivation of a Gauss-type quadrature rule (named as GaussDaubechies quadrature rule) for numerical evaluation of integrals involving product of integrable function and Daubechies scale functions/wavelets. Some of the nodes and weights of the quadrature formula may be complex and appear with their conjugates. This is in contrast with the standard Gauss-type quadrature rule for integrals involving products of integrable functions and non-negative weight functions. This quadrature rule has accuracy as good as the standard Gauss-type quadrature rule and is also found to be stable. The efficacy of the quadrature rule derived here has been tested through some numerical examples.

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