HIGH-ORDER EXPANSION OF THE ENERGY EIGENVALUES OF A RELATIVISTIC COULOMB EQUATION

It is shown that the Herbst equation, i.e., a Coulomb potential wave equation including a free relativistic kinetic energy, presents a rather nontrivial series expansion of its eigenvalues in powers of alpha. In fact, in contrast to Klein-Gordon and Dirac equations, it presents not only odd powers of alpha but also nonanalytic ln alpha terms. The iirst orders (alpha(4), alpha(5)) are obtainable by a standard perturbation method on the Sommerfeld correction. A much more effective and systematic method is proposed to get higher orders (alpha(7)). To appreciate the resulting expansion, we compare it to the known results coming from other approaches to relativistic bound stares, namely Klein-Gordon and Dirac equations in an external field and for the two-body problem, the Breit and Sucher-type equations, and the positronium QED result. (C) 1995 Academic Press, Inc.