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Mathematical Proceedings of the Cambridge Philosophical Society

Scattering from infinite sheets

E. B. Davies

Published Sep 1, 1977• 33 citations
Publisher

Abstract

Abstract We prove the existence and completeness of the wave operators for quantum-mechanical scattering by a potential which does not decrease to zero at infinity in two of the three space directions. We also obtain a new abstract result concerning the continuous dependence of the wave operators on the potential.
DOI
10.1017/s0305004100053962

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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:
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.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
In this paper we give a complete theory of the scattering operator on a rigorous mathematical foundation for a certain class of quantum mechanical systems called, 'simple scattering systems'. These systems are characterized by three properties formulated in Section 3. The relation of these properties to the physical content of scattering is explained. We then proceed to define the wave operator and the scattering operator and prove their existence with mathematically rigorous methods. No assumptions are needed regarding the occurrence of bound states and the location of their energy with respect to the continuum region of the total energy operator. In Section 4 we demonstrate the existence of a certain integral representation of the scattering operator which has often been used as a starting point of the iteration approximation. Certain general properties of scattering systems are obtained in Section 5. This leads to necessary conditions for systems to be scattering systems as well as to a new definition of the scattering operator. The paper concludes with a derivation of the scattering cross-section from the scattering operator. The improvement of this derivation over the older ones sonsists in the special care employed for stating the assumptions which must go into such a derivation and the avoidence of improper functions or other artificial devices.