A. G. Gibson

Transition from time‐dependent to time‐independent multichannel quantum scattering theory

Rigorous proofs are given of several theorems establishing the connection between time‐dependent and time‐independent multichannel scattering theory. The method of proof involves a two‐Hilbert‐space formulation of time‐dependent multichannel theory and the theory of spectral integrals. In particular, the time‐independent theory in the form proposed by Alt, Grassberger, and Sandhas is derived.

N‐body quantum scattering theory in two Hilbert spaces. I. The basic equations

Derivations are given for some transition and resolvent operator equations for multichannel quantum scattering with short‐range potentials. The basic difference between these and previous equations is that the unknown operators act only on the channel subspaces. This is made possible by utilizing, and extending, the two‐Hilbert‐space formulation previously given by the authors [in J. Math. Phys. 14, 1328 (1973)]. The equations in abstract form are of the Lippmann–Schwinger type, differing only in the appearance of certain injection operators from one Hilbert space to the other. When applied to multichannel quantum scattering, the abstract theory yields a new system of equations for the transition and resolvent operators. Uniqueness of the solution to the equations is proved.

Time‐independent multichannel scattering theory for charged particles

Rigorous derivations are given of two time‐independent formulas for the multichannel scattering operator for nonrelativistic charged particle systems. The derivations are based on Dollards time‐dependent theory and use techniques of spectral integration. The formulas involve a complex power of the resolvent operator, in contrast to short‐range formulas. Bilateral Laplace transforms are used to derive a generalized multichannel resolvent equation and to prove existence and uniqueness of the solution. The formulas are applied to recover the well‐known two‐body Coulomb scattering amplitude.

N‐body quantum scattering theory in two Hilbert spaces. II. Some asymptotic limits

Within the framework of two‐Hilbert space scattering theory the existence of the strong Abel limit of a certain operator is proved, leading to the following results. A generalized Lippmann identity is derived that is valid for all channels, rather than only two‐body channels. On shell equivalence of the prior, post and AGS transition operators is rigorously proved, thus closing a gap in previous proofs. Results concerning the existence of the scattering operator as a strong, rather than weak, Abel limit are presented, and their implications with respect to the problem of unitarity are discussed. Finally, the possibility of exploiting operator limits of the Obermann–Wollenberg type is studied, with negative results.

Time‐dependent multichannel Coulomb scattering theory

The formulation by Mulherin and Zinnes of two‐particle Coulomb scattering theory is extended to the multichannel case. The wave operators so obtained are proved by a direct method to be identical with those of Dollard.

Multiparticle scattering theory and the method of coupled reaction channels

Abstract The two-Hilbert-space formulation of multiparticle scattering theory is used to investigate the validity and limitations of the conventional form of the coupled reaction channels (CRC) method. Also a new set of coupled dynamical equations is rigorously derived which includes the conventional CRC equations as a special case.

N‐body quantum scattering theory in two Hilbert spaces. V. Computation strategy

In this paper the development of our previously published theory of approximations for the Chandler–Gibson (CG) equations is continued. In particular, our approximation theory is rigorously brought to the point where N‐particle scattering calculations can begin. This is accomplished by mapping the CG operator equations into a function equation form, where the unknowns belong to a new (third!) computational Hilbert space L. This mapping is facilitated by rescaling the Jacobi momentum variables for the relative free motion of the asymptotic clusters so that surfaces of constant kinetic energy are hyperspheres. The input terms to the resulting equations are expanded in a basis on the surface of the kinetic energy hypersphere. This leads to a system of infinitely many coupled one‐dimensional integral equations with the kinetic energy as the continuous variable. The half‐on‐shell variant of these equations is then transformed to a K‐matrix form. Our approximations result from truncating this system to a finite...

N‐body quantum scattering theory in two Hilbert spaces. IV. Approximate equations

A rigorous mathematical theory of approximations is developed for the time‐independent transition operators of N‐body multichannel nonrelativistic quantum scattering theory. New basic dynamical equations are derived and shown to specify uniquely the approximate time‐independent transition operators. These operator equations represent coupled integral equations with compact kernels, but it is not assumed that the equations that determine the exact transition amplitudes have compact kernels. Convergence of sequences of these approximate time‐independent transition operators to the exact transition operator is established in appropriate limits. Stability of the basic dynamical equations is proved. Resolvent‐type equations and their relation to the limiting absorption principle are investigated. The relation of this theory to the Petryshyn theory of A‐proper operators and to the Feshbach unified theory of nuclear reactions is discussed.

A two-Hilbert-space formulation of multi-channel scattering theory

The time-dependent two-Hilbert-space formulation of the scattering theory is first reviewed and the transition to time-independent theory made (Secs. 1-6). Dynamical equations for the transition operator Tare then written down (Sec. 7). In these equations the cluster-(or channel-) projection operators are incorporated into the transition operator, a feature characteristic of our two-Hilbert-space~formulation. The next step (Sec. 8) is the introduction of a new system of N-body equations, which we call M-operator equations. These equations have a unique solution and are strongly approximation-solvable. The theory of strong approximation-solvability (Sec. 9) with projection type approximations leads to the conclusion that the Tand M-operator equations are also stable. There are other advantages of our equations, when compared (Sec. 10) with, for example, the Faddeev-Yakubovskii equations, that are especially apparent when the number of particles is large but the number of interesting channels is small. Still another advantage is provided by the fact that the solutions to the approximate equations can be related to approximate scattering systems (Sec. 11). This permits insight into the faithfulness with which the approximations represent the physics of the scattering process. The basic development is then brought to a close with a discussion (Secs. 12 and 13) of how to include effects of particle indistinguishability and of Coulomb interactions.

N-body quantum scattering theory in two Hilbert spaces. III. Theory of approximations

Abstract A rigorous mathematical theory of approximations is developed for abstract nonrelativistic quantum scattering systems within the two-Hilbert-space framework. An approximate space of asymptotic states and an approximate asymptotic Hamiltonian must be specified initially. An approximate N -particle Hamiltonian is then constructed and proved to be self-adjoint. Approximate wave operators are shown to exist and, in certain interesting cases, to be asymptotically complete. Certain sequences of the approximate wave operators are proved to converge to the exact wave operators in an appropriate limit. Thus approximate scattering operators are unitary and converge to the exact scattering operator.