Generalized translation operators and the construction of potentials at fixed energy

Abstract

Inverse problems either at fixed angular momentum, or at fixed energy are usually solved by applying generalized translation operators (G.T.O.) [1]. It follows that most of the inverse problems are solvable when a suitable G.T.O. or a variant of it exists. When the energy E is kept fixed such operators exist for zero-spin particle scattering on a nucleus; they exist also in the relativistic case of the Dirac equation. When spin orbit forces are included in nucleon-nucleus scattering, some variant of them can be constructed. In the scattering of a zero-spin particle Newton's method [2] is followed and extended to incorporate complex and Coulomb phase shifts: We have found in both instances the existence of transparent potentials.