A note concerning quantum integrability

Abstract

Heuristic arguments are presented supporting the conjecture that almost all quantum Hamiltonians are integrable in the sense that there exist N(N=number of freedoms) mutually commuting observables (which, in analogy with the classical action variables, can be chosen to be the number operators). This follows from perturbational considerations: the series may converge for almost all perturbations that preserve the discreteness of the spectra, because a 'quantum small denominator' almost always uniformly satisfies the condition of sufficient irrationality. The radius of convergence vanishes if h(cross)=0. The classical limit (as h(cross) to 0) of the quantum integrals of motion generically does not exist.