Inverse Scattering Problem for Dirac Particles. Explicit Expressions for the Values of the Potentials and Their Derivatives at the Origin in Terms of the Scattering and Bound-State Data

Abstract

The inverse scattering problem for the Dirac equation with two central potentials (one a relativistic scalar, the other the fourth component of a 4-vector) is investigated. Exact expressions are obtained for the values of the two potentials and their derivatives at the origin, as a function of the scattering data and bound-state parameters of the particle and the antiparticle for one angular momentum. The consideration is limited to the case of S and P waves (j = ½) and to potentials satisfying a restrictive condition; explicit expressions for the values of the two potentials and their first three derivatives at the origin are given. In the case with only one potential present (the usual Dirac equation), these relationships provide explicit connections between the scattering and bound-state parameters for the particle and those for the antiparticle. For instance, an explicit and simple expression for the binding energy and for the corresponding normalization constant, in terms of the scattering phase shifts of the particle and of the anti-particle, is given in the case with only one bound state, say, for the particle, present (in S or P wave, with j = ½). It is assumed that the two potentials are central, holomorphic, and satisfy the conditions ∫0∞rn|Vi(r)|dr<∞,i=1,2,n=0,1,2.