Construction of Potentials from the Phase Shifts at Fixed Energy

The nonrelativistic potential energy between two spinless particles is deduced from a knowledge of all phase shifts at a given energy. A spherically symmetric potential is found always to exist, but it is not unique. In particular, for every energy, there exists at least one nonzero potential which causes the scattering cross section to be zero. The paper contains both the formal construction procedure and the necessary existence and uniqueness (or lack of it) proofs. Some general examples are included.