Emergence of exponentially small reflected waves

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.

Exponentially accurate semiclassical tunneling wavefunctions in one dimension

“…Analogous results for exponentially small reflected waves when the energy is strictly above a potential bump are presented in [2]. Similar results for non-adiabatic transitions in the Born-Oppenheimer approximation are presented in [8] and [11].…”

“…By the methods of used in [8] and [11], the L 2 norm of the error term induced by r(x, E, ) under the integral sign in (2.19) is of order 3/4+δ e −α(E * ) , for some δ > 0, provided t is large enough. Note that Lemma 1 and Proposition 5 of [2] allow us to get better control of T . (See below.…”

“…The rest of the proof, which consists of showing that χ can be approximated by χ mod and χ ∞ Gauss for different values of x, now relies on Lemma 2.3 and on arguments identical to those in the proof of Theorem 5 of [2].…”

“…order√ .This leads immediately to the following corollary. (See Theorem 6 of[2]. )Corollary 2.6There exist X 0 > 0 and δ > 0, such that for all times t, withX 0 < q t < C −β , we have, in the L 2 sense, χ(x, t, ) = χ Gauss (x, t, ) + O 3/4+δ e −α(E * )/ , where χ Gauss (x, t, ) L 2 = O 3/4 e −α(E * )/ .…”

“…Proof Outline We follow the main steps of the proof of the corresponding result for Theorem 5 of [2], with one notable exception. Since we do not use any superadiabatic representation, the next to leading order term in the asymptotics of c a (x, E, ) is of too low an order to be treated as in [2]. We briefly address this issue in more detail.…”