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 expansions 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 first 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 states, 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.
scientific article; zbMATH DE number 788132