Supergeneralized Runge-Lenz vector in the problem of two Coulomb or Newton centers

Abstract

A so-called two centers problem (TCP) has the following two mathematically equivalent, but physically different embodiments. The first one is the motion of an electron in the field of two stationary Coulomb centers of charges $Z$ and $Z$\ensuremath{'} separated by a distance $R$, which is one of the most fundamental problems in quantum mechanics. The second one is the motion of a planet in the gravitational field of two stationary stars of generally different masses, which is one of the most fundamental problems in celestial mechanics. At least, two groups of authors claimed that for the TCP they derived a supergeneralized Runge-Lenz vector, whose projection on the internuclear (or interstellar) axis is conserved. In the present paper, first, we show that their claims are incorrect: the projection of their supergeneralized Runge-Lenz vector is not conserved. Second, we derive a correct supergeneralized Runge-Lenz vector whose projection on the internuclear or interstellar axis does conserve. Third, since in the literature there are several expressions for the separation constant for the TCP---the expressions not having the form of a projection of any vector on the internuclear or interstellar axis---we provide relations between those expressions and our result.