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

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