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Lecture notes in mathematics

Asymptotic time evolutions for strictly outgoing multiparticle quantum systems with long-range potentials

Volker Enß

Published Jan 1, 1988• 1 citation
Publisher
DOI
10.1007/bfb0100783

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An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
We consider quantum three-body systems interacting via long-range two-body potentials that have arbitrary |x| − e decrease at infinity and a few regularity assumptions (Hormander's potentials). We prove that, in the continuous spectral subspace of the hamiltonian, the states that are orthogonal to all two-cluster channels represent particles which, asymptotically in time: get arbitrarily far separated from each other, are outgoing relative to each other (or incoming if time is reversed). We borrow Hormander's stationary phase estimates, and ideas from Kitada-Yajima for the systematic use of suitable Fourier integral operators in long-range problems. The above results are major steps on the time dependent route proposed by Enss for the proof of three-body asymptotic completeness. The final steps, still for Hormander's potentials, will be given in a subsequent publication On considere des systemes quantiques a 3 corps interagissant par des potentiels de paire a longue portee de decroissance arbitraire |x| − e a l'infini et satisfaisant quelques conditions de regularite locale. On montre que dans le sous-espace spectral continu de l'hamiltonien, les etats qui sont orthogonaux a tous les sous-canaux a 2 corps representent des particules qui, asymptotiquement dans le temps: s'eloignent asymptotiquement 2 a 2; sont 2 a 2 dans un etat «sortant» (ou «entrant» si l'on change le signe du temps). On emprunte les estimations de phase stationnaire de Hormander, et des idees dues a Kitada-Yajima pour l'utilisation systematique dans les problemes a longue portee de certains operateurs integraux de Fourier. Les resultats ci-dessus sont des etapes majeures de la methode «dependant du temps» proposee par Enss pour la preuve de la completude asymptotique a 3 corps