John Gipson

The projection method is a constructive proof of complete integrability of classical many body systems. Roughly, this method consists of establishing a correspondence between a system of n particles interacting via two body forces and free or harmonic motion in a space with many more dimensions. The integration of the equations of motion of the interacting system occurs naturally in the space of higher dimension.

We extend the projection method to the quantum case. We consider four different n body systems where the potential is a sum of two body potentials. The two body potentials are of the form: (I) V(Q) = Q(‘-2), (II) V(Q) = (2sinQ/2)(‘-2), (III) V(Q) = (2sinhQ/2)(‘-2), and (IV) V(Q) = Q(‘-2) + (omega)(‘2) Q(‘2). The wave functions and propagators of these systems are given in the form of integrals involving the wave functions and propagators of free or harmonic motion in spaces of n(‘2) and n(n+1)/2 dimensions. Quantization of angular momentum in the large space yields solutions for discrete values of the coupling constant.