Affiliations: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. E-mail: betz@maths.warwick.ac.uk | Institut Fourier, Université de Grenoble I, BP 74, 38402 St.-Martin-d'Hères, France. E-mail: alain.joye@ujf-grenoble.fr | Mathematisches Institut, Universität Tübingen, Germany. E-mail: stefan.teufel@uni-tuebingen.de
Note: [] Supported by an EPSRC fellowship EP/D07181X/1.
Abstract: We study the time-dependent scattering of a quantum mechanical wave packet at a barrier for energies larger than the barrier height, in the semi-classical regime. More precisely, we are interested in the leading order of the exponentially small scattered part of the wave packet in the semiclassical parameter when the energy density of the incident wave is sharply peaked around some value. We prove that this reflected part has, to leading order, a Gaussian shape centered on the classical trajectory for all times soon after its birth time. We give explicit formulas and rigorous error bounds for the reflected wave for all of these times.
Keywords: above barrier scattering, exponential asymptotics, quantum theory
