Emergence of exponentially small reflected waves

TL;DR

This paper analyzes the quantum scattering of wave packets at a barrier in the semi-classical regime, revealing that the exponentially small reflected waves are Gaussian-shaped and follow classical trajectories with explicit formulas and error bounds.

Contribution

It provides a rigorous analysis and explicit formulas for the exponentially small reflected waves in quantum scattering, including error bounds and Gaussian shape characterization.

Findings

Reflected waves are Gaussian-shaped and centered on classical trajectories.

Explicit formulas and error bounds for the reflected wave are derived.

The analysis applies to energies above the barrier in the semi-classical regime.

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.