<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>N</mml:mi></mml:math>expansions for central potentials revisited in the light of hypervirial and Hellmann-Feynman theorems and the principle of minimal sensitivity

Abstract

The hypervirial and Hellmann-Feynman theorems are used in the methods of $1/N$ expansion to construct Rayleigh-Schr\"odinger perturbation expansion for bound-state energy eigenvalues of spherical symmetric potentials. An iteration procedure of calculating correction terms of arbitrarily high orders is obtained for any kind of $1/N$ expansion. The recurrence formulas for three variants of the $1/N$ expansion are considered in this work, namely, the $1/n$ expansion and the shifted and unshifted $1/N$ expansions which are applied to the Gaussian and Patil potentials. As a result, their credibility could be reliably judged when account is taken of high-order terms of the eigenenergies. It is also found that there is a distinct advantage in using the shifted $1/N$ expansion over the two other versions. However, the shifted $1/N$ expansion diverges for s states and in certain cases is not applicable as far as complicated potentials are concerned. In an effort to solve these problems we have incorporated the principle of minimal sensitivity in the shifted $1/N$ expansion as a first step toward extending the scope of applicability of that technique, and then we have tested the obtained approach to some unfavorable cases of the Patil and Hellmann potentials. The agreement between our numerical calculations and reference data is quite satisfactory.