Pure quantum integrability

Abstract

The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not a 1-1, although some close correspondencies exist. If a classical mechanical system is integrable with invariants that are polynomial in momenta one can construct a corresponding commuting set of differential operators. Here we discuss some 2- or 3-dimensional purely quantum integrable systems (the 1-dimensional counterpart is the Lame equation). That is, we have an integrable potential whose amplitude is not free but rather proportional to $\hbar^2$, and in the classical limit the potential vanishes. Furthermore it turns out that some of these systems actually have N+1 commuting differential operators, connected by a nontrivial algebraic relation. Some of them have been discussed recently by A.P. Veselov et. al.} from the point of view of Baker-Akheizer functions.