On asymptotic perturbation theory for quantum mechanics: Almost invariant subspaces and gauge invariant magnetic perturbation theory

Abstract

Singular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory. Under very general conditions, asymptotic estimations on the Rayleigh-Schrödinger expansions of the perturbed spectral projections are obtained. As a consequence almost invariant subspaces of exponential order are constructed. The results cover practically all singular perturbations considered in nonrelativistic quantum mechanics. In the magnetic field case, under the condition that the magnetic field does not increase at infinity, a gauge invariant perturbation theory leading to convergent series with field-dependent coefficients is developed. © 2002 American Institute of Physics.