Shifted large-Nexpansion for the bound states of the Hellmann potential

The method of shifted large-N expansion, where N is the number of spatial dimensions, is applied to study the bound states of the Hellmann potential, which represents the superposition of the attractive Coulomb potential (-A/r) and the Yukawa potential B exp(-Cr)/r of arbitrary strength B and screening parameter C. It emerges that although the analytic expressions for the energy eigenvalues $E_n$,l yield quite accurate results for a wide range of n,l in the limit of very weak screening, the results become gradually worse as the strength B and the screening coefficient C increase. This happens due to the fact that the effective large-N potential becomes quite shallow in comparison to the true potential and the expansion parameter is not sufficiently small enough to guarantee the convergence of the expansion series for the energy levels. Furthermore, the present analysis reveals an intrinsic limitation of the technique in case of specific superposition of potentials: For certain choices of B, C, n, and l, the structure of the effective potential becomes such that it does not possess a local minimum and consequently the method turns out to be inapplicable to determining the corresponding bound-state energies. However, such a limitation does not persist for a simple screened Coulomb potential and reasonably accurate energy eigenvalues and bound-state normalizations are obtained for the neutral atoms. It is expected that the normalized bound-state wave functions obtained through the shifted large-N formalism may be useful in calculating the oscillator strength, bound-bound dipole transition matrix elements, etc. which have significant importance in atomic processes.