Small Denominators, Frequency Operators, and Lie Transforms for Nearly-Integrable Quantum Spin Systems

Abstract

Based on the previously proposed notions of action operators and of quantum integrability, frequency operators are introduced in a fully quantum-mechanical setting. They are conceptually useful because a new formulation can be given to unitary perturbation theory. When worked out for quantum spin systems, this variant is found to be formally equivalent to canonical perturbation theory applied to nearly-integrable systems consisting of classical spins. In particular, it becomes possible to locate the quantum-mechanical operator-valued equivalent of the frequency denominators which may cause divergence of the classical perturbation series. The results established here link the concept of quantum-mechanical integrability to a technical question, namely the behaviour of specific perturbation series.