Dobrovolska I. Semiclassical approach to the logarithmic perturbation theory for the bound-state problem in quantum mechanics

Objects are the bound-state problem of the Schrodinger, Klein-Gordon and Dirac quantum-mechanical equations, the logarithmic perturbation theory. The aim is the explicit semiclassical treatment of the logarithmic perturbation theory applicable for the calculations the high-order corrections to the eigenenergies and eigenfunctions of the ground and excited bound states of the Schrodinger, Klein-Gordon and Dirac equations. Such analytic and numerical methods have been used: method of asymptotic expansions, methods of the theory functions of complex variables and theory of differential equations. Based upon the Planck's constant expansions and suitable quantization conditions a new procedure for deriving the perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been derived. These formulae provide, in principle, the calculation of the perturbation corrections up to an arbitrary order in the analytic or numerical form. The proposed recursion procedure has been adapted for the use of any renormalization scheme of improving the perturbation expansions obtained. A new analytical method for evaluation of the critical screening parameter of the Debye potential has been found. Spheres of use are the potential models in particle physics, atomic and molecular spectroscopy, problems of solid-state physics.