Time-asymptotic propagation of approximate solutions of Schrödinger equations with both potential and initial condition in Fourier-frequency bands

Florent Dewez (1)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

In this paper, we consider the Schrödinger equation in one space-dimension with potential. We start by proving that this equation is well-posed in H1(R) if the potential belongs to W1,∞(R), and we provide a representation of the solution as a series, called Dyson-Phillips series, by using semigroup theory. We focus our attention on the two first terms of this series: the first term is actually the free wave packet while the second term corresponds to the wave packet resulting from the first interaction between the free solution and the potential. To exhibit propagation features, we suppose that both potential and initial datum are in bounded Fourier-frequency bands; in particular a family of potentials satisfying this hypothesis is constructed for illustration. By representing the two first terms of the series as oscillatory integrals and by applying carefully a stationary phase method, we show that they are time-asymptotically localized in space-time cones depending explicitly on the Fourier-frequency bands. This permits to exhibit dynamic interaction phenomena produced by the potential.

Domaines: Mathématiques [math]

Florent Dewez. Time-asymptotic propagation of approximate solutions of Schrödinger equations with both potential and initial condition in Fourier-frequency bands. 2017. ⟨hal-03142532⟩