Integrability and action operators in quantum Hamiltonian systems - art. no. 056202

For a (classically) integrable quantum-mechanical system with two degrees o f freedom, the functional dependence (H) over cap = H-Q((J) over cap (1),(J ) over cap (2)) of the Hamiltonian operator on the action operators is anal yzed and compared with the corresponding functional relationship H(p(1),q(1 );p(2),q(2)) = H-C(J(1),J(2)) in the classical limit of that system. The fo rmer converges toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, nonuniform. The ex istence of the function (H) over cap = H-Q((J) over cap (1),(J) over cap (2 )) in the integrable regime of a parametric quantum system explains empiric al results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The analysis is carried out for an inte grable one-parameter two-spin model. Additional results presented for the ( integrable) circular billiard model illuminate the same conclusions from a different angle.