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.
Consider a pair of nonrelativistic N-particle (N?2) systems with unitarily equivalent Hamiltonians H and WHW. Typically, WHW 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.
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.
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.
A few-body quantum mechanical system is examined. It is assumed that the scattering operator of the system is a sum of two-body potential operators. Some odd ends in the sum commute. Four-dimensional perturbation potentials and second quantized formalism is used for a solution of the problem. The scattering of two distinct particles in a force field is considered as an example. For this case, the system of the integral equations for the scattering amplitudes are obtained. These amplitudes permit us to construct channel Möller operators. It has been shown that such system has unique solution equivalent to the solution of the corresponding Schrödinger equation.
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.
Given a time-independent Hamiltonian H , one can construct a time-dependent Hamiltonian Ht by means of the gauge transformation Ht=UtHUt†−iUt∂tUt† . Here Ut is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on Ut ensuring that Ht is physical as long as H is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians Ht and H such that the driven many-body dynamics governed by Ht can be reduced to the quench dynamics due to the time-independent H . We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where H=Ht is time-dependent but dynamically integrable.
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2 M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/M U = P / M but not with P, else there is no scattering. Even though H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a Klong K long are seen by an inversely shifted observer as a shifted Klong K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted Kshort K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.