SO(2,1) and strong coulomb coupling. [Klein-Gordon equation, algebra, SO(2,1), scalar product, absorption

Abstract

It is explicitly shown, in the framework of the Klein-Gordon equation, that the algebraic method based upon unitary irreducible representations of the group SO(2,1) used to solve the problem of strong Coulomb coupling (e/sup 2/Z greater than l + /sup 1///sub 2/) is equivalent to constructing solutions that are orthogonal with respect to some mixed scalar product, rather than the standard Klein-Gordon scalar product. This elucidates the difference between the spectra given by Case's method, on the one hand, and the algebraic method, on the other hand. By explicitly computing scattering states, it was further shown that algebraic solutions describe absorption of the particle as in the corresponding classical problem.