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Based on the previously proposed notions of action operators and of quantumintegrability, frequency operators are introduced in a fully quantum-mechanicalsetting. They are conceptually useful because a new formulation can be given tounitary perturbation theory. When worked out for quantum spin systems, thisvariant is found to be formally equivalent to canonical perturbation theoryapplied to nearly-integrable systems consisting of classical spins. Inparticular, it becomes possible to locate the quantum-mechanicaloperator-valued equivalent of the frequency denominators which may causedivergence of the classical perturbation series. The results established herelink the concept of quantum-mechanical integrability to a technical question,namely the behaviour of specific perturbation series.Comment: 17 pages, revtex; to appear in Phys. Rev. A, 1. April 9

Small denominators, frequency operators, and Lie transforms for nearly integrable quantum spin systems