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Journal of Mathematical Physics

Wave operators for long-range hard-core potentials

A. W. Saénz, W. W. Zachary

Published Jun 1, 1976• 1 citation
Publisher

Abstract

In a previous paper, we have proposed a definition of wave operators for hard-core potentials of long range from the standpoint of time-dependent single-channel quantum scattering theory. In the present note, we prove the existence of these operators for a large class of such potentials. The proof involves a combination of methods previously used to establich the existence of wave operators for short-range hard-core potentials and long-range potentials without hard cores.

DOI
10.1063/1.522987

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In a previous paper, we have proposed a definition of wave operators for hard-core potentials of long range from the standpoint of time-dependent single-channel quantum scattering theory. In the present note, we prove the existence of these operators for a large class of such potentials. The proof involves a combination of methods previously used to establich the existence of wave operators for short-range hard-core potentials and long-range potentials without hard cores.
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.
With the aid of a Coulomb ``free propagator'' Uc(±)(t) constructed from the asymptotic form of the conventional time-independent solution to the scattering problem in a pure Coulomb field, Mo/ller wave operators are shown to exist for general Coulomb-like (Coulomb + short-range) potentials. It is emphasized that when scattering occurs in such long-range fields, one should distinguish three classes of states: the prepared and detected states, the scattering states, and certain asymptotic states which are peculiar to long-range potentials. The existence of the third class as a class distinct from the first suggests a connection between Uc(±)(t) and the quantum theory of measurements.
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.
Some years ago, conditions in order that unitarily equivalent Hamiltonians H and W*HW yield the same S operator were investigated by Ekstein and by Coester and his collaborators in the framework of nonrelativistic time-dependent single-channel quantum scattering theory. This subject has turned out to be of considerable importance for nuclear physics, since it constitutes the foundation of a widely used method for constructing phase-equivalent nucleon–nucleon potentials. In the present paper we derive a rigorous and concise necessary and sufficient condition for a pair of unitarily equivalent Hamiltonians, governing the dynamics of a pair of nonrelativistic particles in the center-of-mass system, to yield the same S operator. Our theory, based on a time-dependent approach, applies to very general types of short-range potentials, with or without hard cores, and to an extensive class of long-range potentials. Our necessary and sufficient condition simplifies when certain strong limits W±, related to W, are unitary and when W+=W−. Requirements sufficient for each of these properties to hold are determined. Various examples of operators W such that H and W*HW have the same S operator are discussed.