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International Mathematics Research Notices

Classical and Quantum Scattering for a Class of Long Range Random Potentials

Igor Rodnianski, Wilhelm Schlag

Published Oct 16, 2002• 37 citations
Publisher

Abstract

We prove an almost sure existence of the modified wave operators for a class of Schrödinger operators with random long range potentials. The assumed decay of potential at infinity places it beyond the threshold of the standard class of long range potentials as described in the work of Buslaev-Matveev, Alsholm-Kato, and Hörmander. We develop an approach to quantum scattering which relies on averaging of potential over classical random trajectories. We also establish classical scattering for corresponding classical hamiltonians.

DOI
10.1155/s1073792803201100

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We prove an almost sure existence of the modified wave operators for a class of Schrödinger operators with random long range potentials. The assumed decay of potential at infinity places it beyond the threshold of the standard class of long range potentials as described in the work of Buslaev-Matveev, Alsholm-Kato, and Hörmander. We develop an approach to quantum scattering which relies on averaging of potential over classical random trajectories. We also establish classical scattering for corresponding classical hamiltonians.
In the present paper we shall be concerned with the stationary theory of scattering associated with the Schrodinger operator with a long-range potential. In quantum theory of scattering, many authors have investigated the existence and completeness of wave operators W = s-lim ee~y where J-»±oo HI and H2 are self-adjoint operators acting on a Hilbert space M. Among them, Kato and Kuroda gave an abstract time-independent approach to scattering theory. They derived a stationary form of the wave operator in the following way:
By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. Deceleration makes the threshold between the short-range class of potentials and long range class of potentials 1/(1−λ)1/(1- \lambda)1/(1−λ) for some 0<λ<1/20 < \lambda < 1/20<λ<1/2, where λ\lambdaλ is determined by the mass of particle and coefficients of harmonic potentials. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we prove the existence and completeness of wave operators with respect to such short-range potentials.
We study the asymptotic behaviour of Bohmian trajectories in a scattering situation with short-range potential V and for wavefunctions with a scattering and a bound part. It is shown that the set of possible trajectories splits into trajectories whose long-time behaviour is governed by the scattering part of the wavefunction (scattering trajectories) and trajectories whose long time behaviour is governed by the bound part of the wavefunction (bound trajectories). Furthermore, the scattering trajectories behave like trajectories in classical mechanics in the limit t → ∞. As an intermediate step, we show that the asymptotic velocity exists almost surely and is randomly distributed with the density , where is the outgoing asymptote of the scattering part of the wavefunction.
By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we can prove the existence and completeness of wave operators with respect to the short-range potentials V(t,x)V(t,x)V(t,x) satisfying ∣V(t,x)∣=o((1+∣x∣)−1/(1−λ))|V(t,x)| = o ((1+|x|)^{-1/(1- λ)})∣V(t,x)∣=o((1+∣x∣)−1/(1−λ)), for some λλλ, 0<λ<1/20< λ< 1/20<λ<1/2.