Construction of the Dirac Equation Central Potential from Phase Shifts and Bound States

The connection between phase shifts, bound-state energies, and the potential is studied for a Dirac particle in a spherically symmetric potential. An explicit method is developed for the construction of the potential from the scattering and bound-state data for a single angular momentum and parity. The technique used in this relativistic problem is an extension, appropriately generalized to matrices, of the methods used by Jost and Kohn for the nonrelativistic case. For potentials with $\int 0^{\infty }{r}^n\mathrm{}\left|V\right(r\left)\right|dr$ finite for $n=0, 1, \mathrm{and} 2$, it is shown that a spectral function can be constructed from the phase shifts, the bound-state energies, and the norm of the bound-state wave functions. A generalization of the Gel'fand and Levitan method is developed for the determination of the potential from the spectral function. First, eigenvectors associated with two different potentials are related; from the operator that connects the two systems of eigenfunctions, a modified kernel is defined which satisfies an integral equation determined by the spectral function and eigenfunctions corresponding to a "comparison potential" and the spectral function associated with the "unknown potential." Second, the potential difference is obtained by the differentiation of a certain combination of the elements of the modified kernel.