SLAC - Pm -

A previous work dealing with bound states in a Coulomb potential is extended to the case of scattering states. The symmetry of the problem under the Lorentz group 0(1,3) is used to construct wave functions. Harmonic analysis on two-sheeted hyperboloids is briefly discussed in arbitrary dimension. The set of scattering states for both an attractive and a repulsive potential is shown to provide a unitary representation of the group 0(1,4). (To be submitted to Reviews of Modern Physics) * Work supported by the U. S. Atomic Energy Commission ** On leave from Service de Physique Theorique, CEN Saclay, BP No. 2, Gif-sur-Yvette (Seine et Oise), Prance I. IN!l?RODUCTION In a previous article, hereafter denoted by I, we have reviewed the symmetries of the Schroedinger equation for a Coulomb potential and discussed the use of a non-compact group isomorphic to the pseudo-orthogonal group 0(1,4) to relate the bound state 1evels.l It is of some interest to pursue the analysis further to the scattering states. This is the goal of this work. The "hidden symmetry" is now an invariance under the homogeneous Lorentz group 0(1,3) and the space of scattering states can be written as a direct integral of infinite dimensional Hilbert spaces, which are carrier spaces of unitary representations of the Lorentz group. This is a frequent occurrence when dealing with a non-compact group, the simplest example being Fourier analysis for the group of translations in one dimension. We shall follow the.usual device of introducing non-normalizable scattering states in order to achieve the decomposition. Physically, of course, the direct integral is related to the continuous spectrum of the Hamiltonian. As in the bound state case, there exists a larger group, isomorphic to 0<1,4) which connects the scattering states, but we have to introduce wave functions for both attractive and repulsive potentials. We shall show that the representation of this group, obtained in this way, is equivalent with the one previously discussed in I. This representation is in fact not unique and the ambiguity which arises is the same as the one already encountered. We shall slightly generalize the discussion by taking an arbitrary dimension f for the configuration space. In order to construct the wave functions it will be necessary to use harmonic analysis on a two-sheeted hyperboloid. This has been developed in four dimensions in a series of