An asymptotic analysis of quantum evolution with electromagnetic fields

Abstract

A singular perturbation expansion of solutions to the Schrödinger initial value problem is constructed using an approximate propagator. For a nonrelativistic quantum system interacting with time-dependent external electromagnetic fields, this approximate propagator defines a gauge invariant semiclassical expansion that is realized by large mass scaling. The asymptotic nature of this approximation is established by constructing error estimates that bound the Hilbert space norm difference between the exact and approximate evolved states. The maximum order of the approximation is determined explicitly as a function of the number of derivatives supported by the scalar and vector potentials. The asymptotic expansion is obtained when the configuration space Ω=Rd, and also for problems where Ω is a proper subset of Rd and the self-adjoint Hamiltonian is defined using a supplementary boundary condition—typically Dirichlet or periodic.