Construction of Relativistic Potentials When the Energy Is Fixed

Abstract

We use generalized translation operators for solving the relativistic inverse problem at fixed energy and study the extension of Newton's method. Both the Klein-Gordon equation and the Dirac equation are considered. The determination of the so-called coefficients of interpolation remains a crucial point of the solution. In the case of the Klein-Gordon equation, these coefficients are obtained by the same system of equations as the system obtained for nonrelativistic spinless particles. Therefore, the same singular problem is encountered whether the spinless particles are relativistic or not. In the case of the Dirac equation, the problem with two potentials differs from the problem where only one potential is present. When there are two potentials, a generalized translation operator exists, and the inverse problem can be solved by inverting a singular matrix equation for the coefficients of interpolation. When only one potential is present, the solution of the inverse problem is restricted by a compatibility requirement.