The few-body problem on a lattice

The author explores some of the inherent simplifications of "quantum lattice physics. " He distinguishes between fermions and bosons and analyzes the n-body problem for each, with n=1, 2, 3 typically a small number. With delta-function (zero-range) interactions, the three-body problem on a lattice is manageable, and some results can even be extrapolated to n4. Such calculations are not limited to one dimension (where the well-known Bethe ansatz solves a number of n-body problems). On the contrary, studies cited are mainly in three dimensions and actually simplify with increasing dimensionality. For example, it is found that bound states of n3 particles in d3 dimensions are formed discontinuously as the strength of two-body attractive forces is increased, and are therefore always in the easily analyzed "strong coupling limit. " In the Appendix, an exactly solved example from the theory of itinerant-electron magnetism illustrates how a rigorous solution to the few-body problem is capable of yielding information concerning the N-body problem.