Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Coulomb Scattering. I. Single Channel

Denis Wren Mark Mulherin, I. I. Zinnes

Published Apr 1, 1970• 45 citations
Publisher

Abstract

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.

DOI
10.1063/1.1665275

Related papers

Asymptotic Convergence and the Coulomb Interaction

John D. Dollard

A definition of asymptotic convergence is given for nonrelativistic time-dependent scattering problems involving Coulomb potentials. Convergence proofs have been found both for potential and for n-body multichannel scattering. For pure Coulomb potential scattering, the Mo/ller wave matrix is computed explicitly and found to have its usual meaning.

Journal of Mathematical Physics•Jun 1, 1964•472 citations

Wave operators for long-range hard-core potentials

A. W. Saénz, W. W. Zachary

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.

Journal of Mathematical Physics•Jun 1, 1976•1 citation

Application of the Singular Wave Function Operator in Scattering Theory. II

Robert A. Mapleton

In a recent paper by this author, the singular wave function matrix of Mo/ller was used to obtain the scattering solution for the nonrelativistic Coulomb operator. In the present paper this singular wave function operator is defined differently. This definition is deduced from the relation of the characteristic form of the Green's function to the branch cut in the continuous part of the spectrum of the Green's function. As a consequence of this interpretation, the calculation difficulties of the previous paper do not occur. This method is also used to obtain the scattering solution for the attractive square-well potential.

Journal of Mathematical Physics•Mar 1, 1962•16 citations

Time-dependent scattering theory for infinite delta function potentials in one dimension

John D. Dollard

Existence of the Mo/ller wave operators is proved for a system of n quantum mechanical particles interacting through infinite delta function potentials in one dimension.

Journal of Mathematical Physics•Apr 1, 1978•3 citations

The Quantum-Mechanical Scattering Problem

Thomas F. Jordan

The purpose of this paper is threefold: to provide a mathematically rigorous formulation of the quantum-mechanical scattering problem from the time-independent point of view as has been done by Jauch from the time-dependent point of view, to establish a union between the two formulations, and to investigate the necessity of the asymptotic condition which occurs as a postulate in the time-dependent formulation. The formulation of the problem depends only on the ``total'' and ``free'' Hamiltonian operators. Under the conditions necessary for the time-dependent formulation, the wave operators defined by the asymptotic limits provide a unique solution of this problem. The possibility that solutions can exist when the asymptotic conditions are not valid is investigated by defining wave operators by an integral representation. The conditions sufficient for these to provide a unique solution are shown to be possibly weaker than the asymptotic conditions; there may be a class of Hamiltonian operators for which such solutions exist but for which the asymptotic limits do not. An explicit characterization of such a set of Hamiltonian operators is not achieved, but this question of the necessity of the asymptotic condition has been reduced to a specific mathematical problem. It is hoped that this paper will find a reader who is able to carry the mathematical investigation further.

Journal of Mathematical Physics•May 1, 1962•11 citations

On the Long-Range Stationary Wave Operator

Hiroshi Isozaki

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:

Publications of the Research Institute for Mathematical Sciences•Dec 31, 1977•11 citations
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.
In a recent paper by this author, the singular wave function matrix of Mo/ller was used to obtain the scattering solution for the nonrelativistic Coulomb operator. In the present paper this singular wave function operator is defined differently. This definition is deduced from the relation of the characteristic form of the Green's function to the branch cut in the continuous part of the spectrum of the Green's function. As a consequence of this interpretation, the calculation difficulties of the previous paper do not occur. This method is also used to obtain the scattering solution for the attractive square-well potential.
The purpose of this paper is threefold: to provide a mathematically rigorous formulation of the quantum-mechanical scattering problem from the time-independent point of view as has been done by Jauch from the time-dependent point of view, to establish a union between the two formulations, and to investigate the necessity of the asymptotic condition which occurs as a postulate in the time-dependent formulation. The formulation of the problem depends only on the ``total'' and ``free'' Hamiltonian operators. Under the conditions necessary for the time-dependent formulation, the wave operators defined by the asymptotic limits provide a unique solution of this problem. The possibility that solutions can exist when the asymptotic conditions are not valid is investigated by defining wave operators by an integral representation. The conditions sufficient for these to provide a unique solution are shown to be possibly weaker than the asymptotic conditions; there may be a class of Hamiltonian operators for which such solutions exist but for which the asymptotic limits do not. An explicit characterization of such a set of Hamiltonian operators is not achieved, but this question of the necessity of the asymptotic condition has been reduced to a specific mathematical problem. It is hoped that this paper will find a reader who is able to carry the mathematical investigation further.
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: