Time-asymptotic propagation of approximate solutions of Schr\"odinger equations with both potential and initial condition in Fourier-frequency bands

TL;DR

This paper analyzes the long-term behavior of approximate solutions to the one-dimensional Schrödinger equation with potential and initial conditions confined to specific Fourier-frequency bands, revealing how wave packets evolve and interact over time.

Contribution

It introduces a method to describe the asymptotic propagation of wave packets in Schrödinger equations with band-limited data, highlighting interaction phenomena via space-time cone localization.

Findings

Solutions localize in space-time cones depending on frequency bands

Interaction effects are characterized through asymptotic wave packet behavior

Potential influences are explicitly linked to wave packet dynamics

Abstract

In this paper, we consider the Schr\"odinger equation in one space-dimension with potential and we aim at exhibiting dynamic interaction phenomena produced by the potential. To this end, we focus our attention on the time-asymptotic behaviour of the two first terms of the Dyson-Phillips series, which gives a representation of the solution of the equation according to semigroup theory. The first term is actually the free wave packet while the second term corresponds to the wave packet resulting from a first interaction between the free solution and the potential. In order to follow a method developed in a series of papers and aiming at describing propagation features of wave packets, we suppose that both the potential and the initial datum are in bounded Fourier-frequency bands; in particular a family of potentials satisfying this hypothesis is constructed for illustration. We show then that the two terms are time-asymptotically localised in space-time cones which depend explicitly on the frequency bands. Since the inclination and the width of these cones indicate the time-asymptotic motion and dispersion of the two terms, our approach permits to highlight interaction phenomena produced by the potential.