The semiclassical limit of quantum dynamics

Abstract

We study the ħ→0 limit of the quantum dynamics determined by the Hamiltonian H(ħ) = -(ħ²/2m)Δ + Ｖ on L²(ℝⁿ) for a large class of potentials. By convolving with certain Gaussian states we obtain classically determined asymptotic behavior of the quantum evolution of states of compact support. For suitable potentials we obtain the analogus result for the scattering operator in the position representation. For initial or incoming states of class C_o¹ the error terms are shown to have L² norms of order ħ^½-ε for arbitrarily small positive ε.