On a class of completely transparent nonlocal two-body potentials

Abstract

We show here the existence of a large class of nonlocal two-body potentials which are completely transparent at all energies. These potentials are of the general form given by (3.10),i.e. they are super-positions of a local central potential and a sum of separable potentials. Starting from an arbitrary local potential, we show how to choose the nonlocal separable parts in such a way that the effects of the local potential are cancelled out at all energies and for all angular momenta. The total Hamiltonian therefore becomes equivalent to the free Hamiltonian as long as theS-matrix on the energy shell is concerned. At finite distances, the wave-function is of course different from the free wavefunction, but the difference goes to zero in the limit of large distances. It is perhaps worth mentioning that the transparent potentials we obtain are not pathologica; they have acceptable singularities at the origin and vanish fast at infinity. In the example we study in Sect.4, we have indeed potentials which are less singular thanr−1 at the origin, and decrease exponentially whenr→∞.