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Niveau: Supérieur, Doctorat, Bac+8

Emergence of exponentially small reflected waves Volker Betz? Mathematics Institute, University of Warwick, United Kingdom Alain Joye Institut Fourier, Universite de Grenoble I, BP 74, 38402 St.-Martin-d'Heres, France Stefan Teufel Mathematisches Institut, Universitat Tubingen, Germany November 6, 2008 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. MSC (2000): 41A60; 81Q05. Key words: Above barrier scattering, exponential asymptotics, quantum theory. 1 Introduction We consider the problem of potential scattering for a quantum particle in one dimen- sion in the semiclassical limit. Let V : R ? R be a bounded analytic potential function such that lim|x|?∞ V (x) = 0.

Emergence of exponentially small reflected waves Volker Betz? Mathematics Institute, University of Warwick, United Kingdom Alain Joye Institut Fourier, Universite de Grenoble I, BP 74, 38402 St.-Martin-d'Heres, France Stefan Teufel Mathematisches Institut, Universitat Tubingen, Germany November 6, 2008 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. MSC (2000): 41A60; 81Q05. Key words: Above barrier scattering, exponential asymptotics, quantum theory. 1 Introduction We consider the problem of potential scattering for a quantum particle in one dimen- sion in the semiclassical limit. Let V : R ? R be a bounded analytic potential function such that lim|x|?∞ V (x) = 0.

- semiclassical limit
- dependent decay
- semi-classical regime
- reflected wave
- tunneling can
- exponentially small
- classical reflection probabilities
- smaller than maxx?r

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Publié par | profil-zyak-2012 |

Nombre de visites sur la page | 10 |

Langue | English |

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