A renormalized equation for the three-body system with short-range interactions

We study the three-body system with short-range interactions characterized by an unnaturally large two-body scattering length. We show that the off-shell scattering amplitude is cutoff independent up to power corrections. This allows us to derive an exact renormalization group equation for the three-body force. We also obtain a renormalized equation for the off-shell scattering amplitude. This equation is invariant under discrete scale transformations. The periodicity of the spectrum of bound states originally observed by Efimov is a consequence of this symmetry. The functional dependence of the three-body scattering length on the two-body scattering length can be obtained analytically using the asymptotic solution to the integral equation. An analogous formula for the three-body recombination coefficient is also obtained.     

Coupled channel T-operator equations with connected kernels: (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected (N?1)th iterates of the kernel of the coupled equations is made in the N-channel case (N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.     

Two-Nucleon Scattering Without Partial Waves Using a Momentum Space Argonne V18 Interaction

We test the operator form of the Fourier transform of the Argonne V18 potential by computing selected scattering observables and all Wolfenstein parameters for a variety of energies. These are compared to the GW-DAC database and to partial wave calculations. We represent the interaction and transition operators as expansions in a spin-momentum basis. In this representation the Lippmann–Schwinger equation becomes a six channel integral equation in two variables. Our calculations use different numbers of spin-momentum basis elements to represent the on- and off-shell transition operators. This is because different numbers of independent spin-momentum basis elements are required to expand the on- and off-shell transition operators. The choice of on and off-shell spin-momentum basis elements is made so that the coefficients of the on-shell spin-momentum basis vectors are simply related to the corresponding off-shell coefficients.     

Generalization and applications of the Sasakawa theory

The Sasakawa theory of scattering is phrased in the form of a Fredhohn reduction technique for integral equations possessing a fixed-point singularity in their kernels. This permits the generalization of this theory to a large variety of scattering integral equations. Some specific applications include the two-particle off-shell and multichannel scattering problems. In the first instance a rank-three approximation to the fully off-shell transition matrix is derived which is exact on and half-off shell, satisfies off-shell unitarity, and which possesses no unphysical singularities. In the second problem it is shown how the method leads to the generation of a unitary approximation to the multichannel amplitudes.     

General N-Body Theory of Nonrelativistic Quantum Scattering

 The two-Hilbert-space theory of scattering is reviewed with particular reference to its application to nonrelativistic multichannel quantum- mechanical scattering theory. In Part I the abstract assumptions of the theory are collected, transition operators (both on- and off-energy-shell) are defined, the dynamical equations that determine the off-shell transition operators are presented and their real-energy limits examined, and the convergence of sequences of approximate transition operators is established. A section on how to incorporate group symmetries into the formalism reports new work. The material of Part I is relevant to a variety of both classical and quantum scattering systems. In Part II attention is directed specifically to N-body nonrelativistic quantum scattering systems in which the particles interact via short-range pair potentials. A method of constructing approximate transition operators is presented and shown to satisfy all the abstract assumptions of Part I. The dynamical equations that determine the half-on-shell approximate transition operators are shown to be coupled one-dimensional integral equations that have compact kernels and unique solutions when considered as operators on a Hilbert space of H?lder continuous functions. Moreover, the on-shell parts of those approximate transition amplitudes are shown to converge to the exact on-shell amplitudes as the order of the approximation increases. Detailed formulas for the kernels of the integral equations are written down for systems of particles that are distinguishable and for systems containing identical particles. Finally, some important open problems are described. Received July 2, 1999; accepted in final form October 27, 1999     

Three-body inverse scattering problem

Two methods are suggested to reconstruct three-body potentials from three-body scattering data. This was achieved by using the reduction of the corresponding Schrödinger equation to a system of ordinary differential equations (not integro-differential equations as usual in the direct problem). Exactly solvable three-body models are presented. A new simple method for solving the multi-dimensional inverse problem in a finite-difference approximation is considered in the Appendix.     

Multiple scattering, optical potential, and N-body approaches to elastic two-fragment collisions

Two-fragment elastic scattering problems are often studied using multiple scattering theories such as those due to Watson along with Feshbach-type optical potentials. These conventional methods are re-examined, rephased, and generalized using the language and techniques of contemporary N-particle scattering theory. A special realization of the latter theory is developed which is especially useful for relating the older and newer methods. This is facilitated by maintaining the same off-shell continuations of the scattering operators in both approaches. In particular, a set of connected-kernel scattering integral equations is introduced which provides a consistent N-particle framework for the calculation of that definition of the optical potential possessing the Feshbach off-shell continuation. These equations exhibit a multiple-scattering substructure and therefore allow the systematic generalization of some of the usual low-order approximations.     

The Coulomb integrals and the diffraction model of transfer reactions

The resonating group interaction of three clusters, in the single channel no-distortion approximation, is split into a leading part and a residual part. In norm kernel eigenstate representation, the leading part exhibits a peculiar, fish bone like symmetry. An off-shell transformation, which leads to an energy-independent interaction, also reduces the strength of the three-body Pauli potential. The smallness of this potential is related to the possibility of interpreting cluster relative motion wave functions as probability density amplitudes. Neglecting all residual interactions and introducing, instead, fitting parameters into the two-cluster direct interactions leads to a three-cluster optical model. In this model the on-shell behaviour of two-cluster interactions is determined by experimental data, while their off-shell behaviour, as well as the three-cluster Pauli potential, are determined by the Pauli principle.     

A non-singular equation for the three-body scattering problem

This paper derives a non-singular integral equation for the three-body problem. Starting from the three-body equations obtained by Karlsson and Zeiger we introduce a set of algebraic transformations that remove all the Green function pole singularities. For scattering energies on the real axis we find a singularity-free momentum-space integral equation. This equation requires only a finite range of momentum values for its solution. In the case of well-behaved two-body interactions, such as the superposition of Yukawa interactions, we prove that the kernels of this equation have a finite Hilbert-Schmidt norm. This same norm provides a general criteria for establishing when the impulse approximation is accurate.     

Elastic pion-deuteron scattering in the resonance region as a three-body problem

We study elastic pion-deuteron scattering in the Δ(1236) energy region by means of the three-body Faddeev equations. We present a compact angular momentum reduction of the Faddeev integral equation for separable amplitudes, neglecting the nucleon spin, and solve the resulting coupled integral equations. We examine the dependence of the elastic scattering amplitude on the deuteron structure, on the pion-nucleon scattering amplitude, and on the various orders of multiple scattering. The differential cross section is very sensitive to multiple scattering effects at backward angles. We find that a number of conventional approximations do not well reproduce these multiple scattering effects in the resonance region.     

A new method in the theory of many-body scattering

New coupled equations for the transition (T) and reactance (K) operators for N-channel, many-body, rearrangement scattering are derived. The key to the new method is the channel coupling array W, which links the various rearrangement channels together. By specializing W to the class of channel permuting arrays, it is shown that the (N−1)st iterate of the kernel of the coupled equations is connected.     

Two-body scattering without angular-momentum decomposition: Fully off-shell T-matrices

Two-body scattering is studied by solving the Lippmann-Schwinger equation in momentum space without angular-momentum decomposition for a local short-range interaction plus Coulomb. The screening and renormalization approach is employed to treat the Coulomb interaction. Benchmark calculations are performed by comparing our procedure with a configuration space calculation, using the standard partial-wave decomposition, for ¹²C - ¹⁰Be elastic scattering. The fully off-shell T -matrices are also calculated for the final goal of studying the three-body scattering by solving Faddeev/AGS equations.     

Kernel approximation for solving few-body integral equations

This paper investigates an approximate method for solving integral equations that arise in few-body problems. The method is to replace the kernel by a degenerate kernel defined on a finite dimensional subspace of piecewise Lagrange polynomials. Numerical accuracy of the method is tested by solving the two-body Lippmann-Schwinger equation with non-separable potentials, and the three-body Amado-Lovelace equation with separable two-body potentials.     

Theory of elastic π3He scattering

The pion-three-nucleon system is investigated using coupled Schrödinger equations. The coupling between the four-body (πNNN) and three-body (NNN) systems is explicitly implemented by operators for emission and absorption of the pion by each nucleon. The only simplifying assumption is the separable form for amplitudes pertaining to pure potential scattering. A set of Amado-Lovelace type equations is derived, from which the amplitude for the reaction π + ³He→ π + ³He can be evaluated. The integral equations involve intermediate integration over single relative momenta so that subsequent numerical solution is within reach.     

A unitary model for three-body collisions

A connected 3 → 3 formalism for three-body collision processes is reduced to a hierarchy of three on-energy-shell integral equations and one off-energy-shell integral equation. Only the on-energy-shell equations, which involve only on-energy-shell three-body and two-body amplitudes, need be solved exactly in order to obtain elastic and break-up amplitudes satisfying the unitarity constraints exactly. Applied to n-d break-up, the on-energy-shell equations ensure that the n-d initial-state interaction, the nucleon-nucleon final-state interactions, and more complicated 3 → 3 processes are correctly described. After angular momentum analysis the on-energy-shell equations are one-dimensional integral equations, even in the case of local two-body potentials. This unitary model provides a practical scheme for calculating approximate three-body elastic and break-up amplitudes when two-body local potentials are used to describe the two-body subsystems.     

Use of wavelets in potential scattering problems

Application of the Daubechies compact support wavelets to problems of nonrelativistic potential scattering described by integral Lippmann-Schwinger equation is discussed. Structure of wavelet representation of various physical operators is investigated. It is shown that for a special class of potentials wavelets enable sparse approximation of the kernel of the Lippmann-Schwinger equation. Constraints for such potentials are derived. This paper is dedicated to Prof. J. Bičák on the occasion of his 60th birthday.     

Non-orthogonality and overcompleteness in direct nuclear reactions

When rearrangement reactions are too strong to be treated by a one-step DWBA, they must be described by coupled reaction channel (CRC) equations or by a multistep DWBA derived from them. The derivation of these equations requires the use of projection operators on subspaces which are in general not orthogonal and which may be linearly dependent. We consider the case of many coupled two-cluster channels, and show how solving the non-orthogonality kernel to give an orthogonalized CRC formally simplifies the structure of the equation and clarifies the relations between different methods, including connected kernel approaches. We use the requirement that the distorted Faddeev (N = 3) or LBRS equations (general N) be satisfied in the two-cluster model space. We demonstrate that this determines the distortion potentials and that the resulting pole approximation yields the orthogonalized CRC equations. A modified one- and two-step DWBA is written in which non-orthogonality corrections are summed to all orders in each step. Methods of generating the non-orthogonality correction operator are discussed.     

-Body Quantum Scattering Theory in Two Hilbert Spaces: -Body Integral Equations

Scattering for a nonrelativistic system of distinguishable and spinless particles interacting via short-range pair potentials is considered. Half-on-shell integral equations (the CG equations) are proposed, the solutions of which determine approximate scattering amplitudes that converge to the exact scattering amplitude. It is proved, under mild H?lder integrability assumptions, that these apparently singular equations actually have a compact kernel for real energies and, consequently, a unique solution. The CG equations have a structure that is much simpler than the Yakubovskii equations and similar to that of coupled-reaction-channel equations. The driving terms look like distorted-wave Born integrals and nonorthogonality integrals. However, there is no restriction to channels with only two asymptotic bound clusters and for all channels, no matter how many bound clusters, appropriate boundary conditions are exactly satisfied. This work completes the establishment of a rigorous mathematical link between the solutions of the half-on-shell CG equations and the on-shell transition operators defined in time-dependent multichannel scattering theory, and it provides for the first time a rigorous theoretical basis for practical calculations of scattering amplitudes for certain problems with . Received October 27, 1997; accepted for publication December 29, 1997     

A resolution of some orthogonality and expansion-set problems in many-body scattering theory

A modification to the coupled channel equations for many-body scattering is introduced, based on the evaluation of the discontinuity equation for the matrix of transition operators. The resulting (modified) equations are shown to be in a form which resolves some problems associated with continuum states in an analysis of rearrangement collisions using the eigenstate expansion method.