Noncommutative Adiabatic Phase

In the problem of separation of "fast" variables and "slow" parameters for quantum systems the application of the general operator scheme of the Hamiltonian adiabatic approximation is developed. Formulas for adiabatic connection and curvature for Hamiltonians represented as a quadratic form with respect to fast variables, as well as the formula for the effective Hamiltonian in the first (after zero) adiabatic approximation are found. The present work touches upon the simplest models of one-dimensional and two-dimensional harmonic oscillators parametrized by "slow" non-commuting coordinates on a two-dimensional space. For these models the explicit formulas for connection and curvature are obtained. Moreover, an explicit representation of the effective Hamiltonian in the first (after zero) adiabatic approximation is found. The commutation relations between the generators of the algebra of "fast" symmetries and the "slow" coordinates of the guiding center are given.