Scalar particle with Darwin-Cox structure in external Coulomb field

Abstract

Generalized Klein-Fock-Gordon equation for a scalar particle with Darwin-Cox structure, which takes into account distribution of electric charge of the particle inside a finite spherical region is studied in external Coulomb field. Corresponding radial equation has two irregular singular points, r = 0 of the rank 3, r = ∞ of the rank 2, and four regular singular points. In the case of minimal angular momentum, l = 0, the structure of singularities becomes simpler: the points r = 0, r = ∞ are both of the rank 2, and four regular points remain the same. There are constructed formally exact Frobenius type solutions of the derived equations, convergence of relevant power series, with 8-term and 7-term recurrent relations respectively, is studied. As analytical quantization rule is taken so-called transcendence conditions. It provides us with 4-th order algebraic equation with respect to energy values, which has four sets of roots. Only one set of roots, 0 < El,k < 1, depending on angular momentum l = 0, 1, 2, … and main quantum number n = 0, 1, 2, … may be interpreted as corresponding to some bound states of the particle in the Coulomb field. In the same manner, a generalized nonrelativistic Schrödinger equation for such a particle was studied, the final results are similar.