Quasipotential Equation Corresponding to the Relativistic Eikonal Approximation

Abstract

A three-dimensional Lippmann-Schwinger-type equation for the elastic scattering amplitude and the corresponding homogeneous Schr\"odinger equation for the two-particle bound states are studied. The potential is defined as an infinite power series in the coupling constant which fits the perturbative expansion of the on-energy-shell scattering amplitude. The approximate equation obtained by keeping only the lowest-order term in the potential is local and has the following properties: (i) The scattering amplitude yields the relativistic eikonal approximation for large energies or small exchanged mass and momentum transfer; (ii) for the Coulomb problem the approximate equation is exactly soluble and leads to a relativistic Balmer formula including the fine-structure splitting.