Emergence of exponentially small reflected waves

Article type: Research Article

Authors: Betz, Volker^; | Joye, Alain | Teufel, Stefan

Affiliations: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. E-mail: [email protected] | Institut Fourier, Université de Grenoble I, BP 74, 38402 St.-Martin-d'Hères, France. E-mail: [email protected] | Mathematisches Institut, Universität Tübingen, Germany. E-mail: [email protected]

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

DOI: 10.3233/ASY-2009-0935

Journal: Asymptotic Analysis, vol. 64, no. 1-2, pp. 53-100, 2009