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

Physical Review

Equivalent Hamiltonians in Scattering Theory

H. Ekstein

Published Mar 15, 1960• 65 citations
Publisher

Abstract

This paper is a contribution to the discussion of the question: to what extent does the scattering matrix determine the Hamiltonian? The Hamiltonians considered are nonrelativistic, but in extension of previous studies, "nonlocal" potentials and many-body potentials are allowed. A large class of unitary transformations is found which produce Hamiltonians leading to the same $S$ matrix. In the last section, it is shown that this equivalence is only a special consequence of the general axiomatic formulation of scattering in field theory.

DOI
10.1103/physrev.117.1590

Related papers

Unitarily equivalent multiparticle Hamiltonian systems yielding equal scattering for corresponding states

A. W. Saénz, W. W. Zachary

Consider a pair of nonrelativistic N-particle (N?2) systems with unitarily equivalent Hamiltonians H and W*HW. Typically, W*HW involves nonlocal multiparticle interactions even when only local pair interactions are present in H. For the respective cases when H involves short-range interactions alone and also when it involves long-range ones, sufficient conditions are established in order for a scattering amplitude of the first system pertaining to any states in given entry and exit channels to equal the amplitude of the second system pertaining to corresponding states. This is accomplished by a time-dependent approach. The correspondence in question assigns to each channel state of the first system a channel state of the second system in a bijective and intuitively natural manner. Nontrivial examples are given of unitary operators W for which the above equality holds for all channels of these systems. This applies to a large class of interactions in H, including interactions with suitable long-range parts. The present work is the theoretical foundation of a new method of the authors, discussed elsewhere, for investigating many-body nuclear forces phenomenologically.

Journal of Mathematical Physics•Aug 1, 1978•2 citations

Path-integral formulation of scattering theory: Central potentials

Christopher C. Gerry, Vijay A. Singh

We consider central-potential scattering and determine a path-integral representation for the $S$ matrix in polar coordinates. This is obtained by transforming to polar coordinates a Cartesian form of the nonrelativistic $S$ matrix given by Campbell et al., and implementing an idea of Faddeev to obtain the appropriate asymptotic conditions. Our results are applied to scattering in an inverse-square potential to determine the correct phase shifts as well as the $S$ matrix.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•May 15, 1980•11 citations

On the equality of S operators corresponding to unitarily equivalent Hamiltonians in single channel scattering

A. W. Saénz, W. W. Zachary

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.

Journal of Mathematical Physics•Mar 1, 1976•6 citations

Path-integral formulation of scattering theory

William B. Campbell, P. Finkler, C. E. Jones +1 more

A new formulation of nonrelativistic scattering theory is developed which expresses the $S$ matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the $S$ matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving $\ensuremath{\delta}$ function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path-integral expression for the $T$ matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Oct 15, 1975•29 citations

Rotational Invariance and the S Matrix in Nonrelativistic Quantum Mechanics

R. Fong, Joseph Sucher

The connection between an observable invariance property of the scattering matrix and the interaction, $V$, generating the scattering, is studied for a particlar case: rotational invariance in nonrelativistic "potential" scattering. It is shown that a determination that the scattering cross section depends only on the angle between the incident and outgoing beams by no means implies that $V$ is invariant under rotation. This is true even if coherent incident beams are used to probe the target. Some aspects of these results and their relevance for the construction of physical theories are briefly discussed.

Physical Review•Mar 15, 1963•0 citations

Dispersion Relation for Nonrelativistic Particles

David Y. Wong

It is shown that if the wave function of a nonrelativistic particle satisfies the Schr\"odinger equation with a velocity-independent potential, then its scattering amplitudes (and the $S$ matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the $S$ matrix, the complete $S$ matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the $S$ matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.

Physical Review•Jul 1, 1957•11 citations
This paper is a contribution to the discussion of the question: to what extent does the scattering matrix determine the Hamiltonian? The Hamiltonians considered are nonrelativistic, but in extension of previous studies, "nonlocal" potentials and many-body potentials are allowed. A large class of unitary transformations is found which produce Hamiltonians leading to the same SSS matrix. In the last section, it is shown that this equivalence is only a special consequence of the general axiomatic formulation of scattering in field theory.
Consider a pair of nonrelativistic N-particle (N?2) systems with unitarily equivalent Hamiltonians H and W*HW. Typically, W*HW involves nonlocal multiparticle interactions even when only local pair interactions are present in H. For the respective cases when H involves short-range interactions alone and also when it involves long-range ones, sufficient conditions are established in order for a scattering amplitude of the first system pertaining to any states in given entry and exit channels to equal the amplitude of the second system pertaining to corresponding states. This is accomplished by a time-dependent approach. The correspondence in question assigns to each channel state of the first system a channel state of the second system in a bijective and intuitively natural manner. Nontrivial examples are given of unitary operators W for which the above equality holds for all channels of these systems. This applies to a large class of interactions in H, including interactions with suitable long-range parts. The present work is the theoretical foundation of a new method of the authors, discussed elsewhere, for investigating many-body nuclear forces phenomenologically.
We consider central-potential scattering and determine a path-integral representation for the SSS matrix in polar coordinates. This is obtained by transforming to polar coordinates a Cartesian form of the nonrelativistic SSS matrix given by Campbell et al., and implementing an idea of Faddeev to obtain the appropriate asymptotic conditions. Our results are applied to scattering in an inverse-square potential to determine the correct phase shifts as well as the SSS matrix.
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.
A new formulation of nonrelativistic scattering theory is developed which expresses the SSS matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the SSS matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving $\ensuremath{\delta}$ function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path-integral expression for the TTT matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.
The connection between an observable invariance property of the scattering matrix and the interaction, VVV, generating the scattering, is studied for a particlar case: rotational invariance in nonrelativistic "potential" scattering. It is shown that a determination that the scattering cross section depends only on the angle between the incident and outgoing beams by no means implies that VVV is invariant under rotation. This is true even if coherent incident beams are used to probe the target. Some aspects of these results and their relevance for the construction of physical theories are briefly discussed.
It is shown that if the wave function of a nonrelativistic particle satisfies the Schr\"odinger equation with a velocity-independent potential, then its scattering amplitudes (and the SSS matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the SSS matrix, the complete SSS matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the SSS matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.