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 $n=1,2,3\mathrm{}\dots$ 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 $n\ge 4$. 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 $n\ge 3$ particles in $d\ge 3$ 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.