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The derivation and interpretation of the many-particle Lippmann-Schwinger (NS) equation are reviewed. The derivation from the Schrodinger equation involves certain surface integrals at infinity, originally examined by Gerjuoy. Reasonable assumptions about the values of these surface integrals at real energies imply that the many-particle LS equation generally has nonunique solutions. Various arguments, including calculations in a model problem, indicate that these surface integrals have been correctly evaluated, despite recent claims to the contrary.

We compare the different theoretical approaches to the study of multistep compound reactions and establish new connections among them. It is established that a new approach by the present author and that by Feshbach, Kerman and Koonin can be viewed as the eigenclass representation of the approach by Agassi, Weidenmuller and Mantzouranis. Hence the approach by the present author is shown to unify the approaches by Friedman, Hussein, McVoy and Mello and by Feshbach, Kerman and Koonin.

Compound nuclear processes should in fact exhibit all the resonances of the underlying compound nucleus. In intermediate resolution experiments where the energy resolution $\ensuremath{\Delta}E$ is considerably large compared to the width and spacing of compound nuclear states but small compared to those of optical model shape resonances, one observes the well-known intermediate structure. In the case of compound elastic processes a dynamically reasonable account of intermediate structure resonances was developed by various authors, where one assumes the existence of a doorway state through which the incident (or the final) state couples to the compound nuclear states. An energy average of the resultant amplitude exhibits intermediate structure. In the case of compound inelastic reactions the incident and the final states are different and the doorway states that relate the final state to the compound nuclear states will in general be different from the doorway states which relate the incident state to the compound nuclear states. The former doorway states will be referred to as the exit doorway states and the latter as the entrance doorway states. A dynamical theory is developed acknowledging the notion of these two kinds of doorway states. The energy averaged transition amplitude shows two sets of intermediate structure resonances corresponding to these two kinds of doorway states. It is expected that one of these sets of resonances will usually dominate a reaction in an energy domain. The present formulation can possibly explain the different sets of intermediate structure resonances observed in different exit channels of a nuclear reaction, for example, those observed in various exit channels of the $^{12}\mathrm{C}$+$^{16}\mathrm{O}$ system.

Contrary to recent claims we show in a soluble three-body model that the exact scattering wavefunction with incoming wave in a rearrangement channel satisfies the homogeneous Lippmann-Schwinger equation, and hence that the solutions to the Lippmann-Schwinger equation are nonunique.

Made available in DSpace on 2022-04-28T18:55:40Z (GMT). No. of bitstreams: 0 Previous issue date: 1987-01-01 Both the variational approach and the method of separable expansion have proved to be very useful for the solution of scattering equations. Connections are established between these two approaches. It is shown that an underlying variational principle is responsible for the good convergence obtained with the method of separable expansions. It is pointed out that a proper choice of expansion functions is essential for the rapid convergence of the method of separable expansion. Several types of separable expansions are tested numerically for the Reid S01 soft-core potential. It is shown that with proper choices of expansion functions these separable expansions yield more rapid convergence than those obtained previously in other studies of separable expansions. In the case of the three-body scattering equations the usual separable expansions are shown to follow from the general expansion schemes studied here with special choices of expansion functions. © 1987 The American Physical Society. Departamento de Fsica, Universidade Federal de Pernambuco, 50.000 Recife, Pernambuco Instituto de Fsica Terica, Universidade Estadual Paulista, 01.405 So Paulo, So Paulo Instituto de Fsica Terica, Universidade Estadual Paulista, 01.405 So Paulo, So Paulo

A recently proposed separable expansion of the $t$ matrix (assuming the potential to be given) is rederived from the Schwinger variational principle. Also, analytic expressions are given for the integrals in the separable expansion if the potential is a superposition of Yukawa potentials, and for suitable choices of the expansion functions. Calculations with two commonly used $S$-wave nucleon-nucleon potentials show that the expansion can converge significantly faster than the unitary pole expansion if the freedom in the choice of the expansion functions is exploited.

Finite rank expansions for two- and three-body t matrices are analytically continued to the unphysical sheet of the complex energy plane associated with the lowest two-body scattering threshold in order to obtain the position and residue of the virtual state and resonance poles. The present method is applied to study the S01 virtual state of two nucleons, the Efimov virtual states of three identical bosons, and the doublet virtual state of three nucleons. NUCLEAR REACTIONS Separable expansions for t matrices, resonances and virtual states, virtual states for two and three nucleons, Efimov virtual states.

The dramatic failure of the effective range expansion for systems with small scattering length a (| a | ∼ 0) is accounted for theoretically. An alternate expansion is suggested for such systems and is applied successfully to the S wave neutron-deuteron and pion-pion systems where the usual expansion fails because of the appearance of sub-threshold poles in the effective range function.

This paper is concerned with understanding the physics responsible for the major result of the preceding article, viz., that the coupled reaction channel (CRC) ansatz responds better than any of the connected-kernel (exact) collision theories to an L 2 (pseudostate) discretization approximation. The analysis is carried out within the framework of the wave function equations equivalent to the transition operator formalisms employed in the preceding article. In an alternate approach comparing the CRC method with an approximate version of a multichannel collision theory, it was argued that since only the CRC ansatz is based directly on the Schrodinger equation and on orderly restrictions of the Hilbert space, then it must be the superior approximation. While this conclusion is correct, the argument underlying it is in fact inadequate. The different argument used in the present analysis is based on the fact that connected-kernel theories require contributions from the full three-body breakup continua in order to determine the two-body transition amplitudes accurately, while the effect of an L 2 discretization approximation on them is to eliminate much of these contributions unless very high-lying pseudostates are employed. In contrast, high-lying pseudostates are much less important for the CRC case. This is demonstrated and its ramifications are explored in some detail herein. In particular, the analyses based on this argument are seen to yield an understanding and interpretation of almost all of the trends exhibited in the results tabulated in the preceding article.

From a comparison of experimental results with various theoretical calculations of N-N spin transfer parameters in N-d scattering, we find a very strong correlation between the N-d doublet scattering length and these spin transfer parameters, in view of which, it is extremely unlikely that useful information about the N-N interaction can be obtained from these parameters which is not already implicit in the value of the N-d doublet scattering length. In addition, these spin parameters carry some information about the noncentral part of the N-N interaction, which can be conveniently found and studied through the consideration of a new function ..delta..K of these spin parameters.

The aim of this work is to show how we can improve systematically the rate of convergence of a recently proposed method for iterative solution of scattering integral equations. The method relies on the introduction of an auxiliary equation containing an arbitrary function, whose kernel is much weaker than that of the original equation. The solution of the original equation is then expressed in terms of that of the auxiliary equation which is supposed to have a convergent iterative solution. In this work we introduce successive subtractions in the kernel in order to make it weaker. Such subtractions in the kernel are very simple to implement in practice and the present method appears to have some advantage over the other existing methods. We explain how to generalize the method for multichannel scattering equations. Using the present method we show how to modify a method by Fuda for three-particle scattering equations in order to have a method which uses the iterative solution only of nonsingular equations above the three-body breakup threshold. The method is used numerically to compute phase shifts, scattering lengths, and fully off-shell $t$ matrix elements for neutron-deuteron scattering in the $s$-wave Amado model and for nucleon-nucleon scattering with the Reid $^{1}S_{0}$ potential. The iterative solution of the auxiliary equation is found to converge much faster than the conventional Pad\'e technique in the case of the neutron-deuteron scattering and yields results with high precision.[NUCLEAR REACTIONS Multichannel scattering equations, iterative solution of auxiliary equations, Reid $^{1}S_{0}$ potential, Amado model, off-shell $t$ matrix elements and phase shifts computed.]

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind is extended to singular equations of the Lippmann-Schwinger type. Numerical calculations of on- and off-shell t-matrix elements are carried out at positive and negative energies, for the 1S0 Reid soft-core nucleon-nucleon potential. Satisfactory convergence is achieved, even with the simplest version of the s

The constraints of unitarity and analyticity on four-body final state amplitudes are studied. It is shown that unitarity alone forces the amplitudes to be coherent and have rapid (singular) behavior. The implementation of unitarity yields a set of linear integral equations for the four-body amplitudes that are the minimal set consistent with quantum mechanics, but are also equivalent to the full dynamical equation with separable interactions.

A simple connection is established between a recently proposed method of continued fractions for on- and off-shell t matrix elements of local and nonlocal potentials and a more conventional iteration-subtraction method for the same problem. We compare the numerical convergence rates of these methods and the method of Pad\'e approximants for the case of two nucleons interacting via the $^{1}\mathrm{S}_{0}$ Reid soft core potential.

In this paper, the problems of nonorthogonality and possible overcompleteness in the resonating group method equations are treated by explicitly symmetrizing previously developed distinguishable-particle methods. It is shown that the orthogonalized resonating group method equations are equivalent to the two-cluster-subspace projections of a symmetrized set of multiparticle scattering equations with distortion operators. The result of the analysis is the conversion of the resonating group method to an in-principle, convergeable approximation scheme. Corrections to the resonating group method from three-cluster contributions are briefly discussed.

If two particles are interacting via a short range potential and a repulsive Coulomb potential the t matrix can be written as a sum of the Coulomb and the ''nuclear'' t matrices. In order to solve the three-nucleon problem with Coulomb interactions usually we need a separable representation of this ''nuclear'' t matrix. A recently proposed method for finding a separable expansion for local potentials is here extended to find a rapidly convergent separable expansion, with analytic form factors, for the ''nuclear'' part of the t matrix of a local potential, in the presence of Coulomb interactions. The method is illustrated for a two-term Malfliet-Tjon potential. In each rank the ''nuclear'' phase shift is close to the corresponding phase shift when the Coulomb interaction is switched off. (AIP)

A theory of nuclear reactions and structures is proposed which unifies and generalizes several different approaches, such as the method of coupled reaction channels and the distorted wave Born approximation for rearrangement processes, the method of coupled channels array, the resonating group method, the Feshbach theory for the optical potential, and the multiple scattering formalism for high energy processes. The present theory employs an optical potential which can be systematically improved in a controlled way.

An alternate representation of the three-body scattering equations of Alt, Grassberger, and Sandhas is suggested. Like the formulation of Karlsson and Zeiger these equations require only two-body bound-state wave functions and half-off-shell transition amplitudes as input and contain three-body energy-independent effective potentials which become real after partial-wave decomposition. It is emphasized that such representations are particularly suitable for writing singularity-free momentum-space integral equations in the scattering region. One scheme for writing such equations is discussed.

A new class of antisymmetrized, two-fragment, elastic scattering optical potentials having no elastic unitary cuts is constructed. The construction is based on the wave function equations of many-particle scattering theory and yields potentials differing significantly from those derived recently.

The constraints of unitarity and analyticity on four-body final states are studied. It is shown that unitarity alone forces the amplitudes to be coherent and have singular behaviour. The implementation of unitarity with total energy analyticity yields a set of relativistic linear integral equations for the four-body amplitude. This is the minimal set consistent with quantum mechanics and also is the full dynamical set of equations with two-body separable interactions. These equations will provide important ingredients for the phenomenological analysis of four-body final states using the isobar model.

The possibility of a zero-energy Efimov effect is investigated in a model consisting of three identical heavy particles and a lighter one, when the light-heavy interaction leads to a zero-energy bound state for the two-heavy---one-light subsystem. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable $s$-wave potential of Yamaguchi form. The heavy-heavy interaction is short range and, if attractive, is to be taken weak enough to support no two- or three-heavy-particle bound states. The relevant parameter is the potential strength $\ensuremath{\lambda}$ of the light-heavy interaction which for $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ supports a single zero-energy light-heavy bound state. If the first zero-energy bound state of the two-heavy---one-light subsystem occurs for $\ensuremath{\lambda}={\ensuremath{\lambda}}^{\ensuremath{'}}$ such that $|{\ensuremath{\lambda}}^{\ensuremath{'}}|l|{\ensuremath{\lambda}}_{c}|$, there are no Efimov four-body bound states. On the contrary, if the chosen heavy-heavy potential is repulsive enough to prevent the existence of three-body bound states for $|\ensuremath{\lambda}|l|{\ensuremath{\lambda}}_{c}|$, then as $\ensuremath{\lambda}\ensuremath{\rightarrow}{\ensuremath{\lambda}}_{c}$ a few four-body Efimov states may emerge but their number remains finite. These states disappear for $|\ensuremath{\lambda}|\ensuremath{\gtrsim}|{\ensuremath{\lambda}}_{c}|$ as the three-body cut overrides them.

A bound-state approximation when applied to certain operators, such as the many-body resolvent operator for a two-body fragmentation channel, in many-body scattering equations, reduces such equations to equivalent two-body scattering equations which are supposed to provide a good description of the underlying physical process. In this paper we test several variants of bound-state approximations in the soluble three-boson Amado model and find that such approximations lead to weak and unacceptable kernels for the equivalent two-body scattering equations and hence to a poor description of the underlying many-body process.

Like three-body amplitudes, four-body amplitudes have subenergy threshold singularities over and above total-energy singularities. In the four-body problem we encounter a new type of subenergy singularity besides the usual two- and three-body subenergy threshold singularities. This singularity will be referred to as ''independent-pair threshold singularity'' and involves pair-subenergy threshold singularities in each of the two independent pair subenergies in four-body final states. We also study the particularly interesting case of resonant two- and three-body interactions in the four-body isobar model and the rapid (singular) dependence of the isobar amplitudes they generate in the four-body phase space. All these singularities are analyzed in the multiple-scattering formalism and it is shown that they arise from the ''next-to-last'' rescattering and hence may be represented correctly by an approximate amplitude which has that rescattering.

We generalize a simple version of the optical background representation for the transition amplitude to the case of well-nested doorways. Using this simple generalization of the optical background representation an alternate statistical theory of multistep compound reactions is suggested, which uses the idea of nested averaging in the case of well-nested doorways and is valid under very general conditions. Simple connections are established between the present approach of multistep compound reactions and that of Feshbach, Kerman, and Koonin, of Agassi, Weidenm\"uller, and Mantzouranis, and of Hussein and McVoy. The present formulation exhibits the chaining condition of Feshbach, Kerman, and Koonin in the extreme low energy limit, is time reversal symmetric, and yields the Bohr description of compound nuclei in the appropriate limit.

The constraints of unitarity and analyticity on four-body final state amplitudes are studied in the quasi-two-body scheme. The implementaton of unitarity with total energy analyticity yields the minimal set of scattering equations for the problem consistent with constraints of quantum mechanics. The minimal set we obtain gives a dynamical scheme which is distinct from the full four-body scattering scheme. Nevertheless, with the assumption of separable approximation for two- and three-body interactions we get simple Lippmann-Schwinger type equations for four identical bosons for the following two-to-two processes: nt ..-->.. nt, nt ..-->.. dd, dd ..-->.. dd, and dd ..-->.. nd. Here n, d, and t refer to nucleon, deuteron, and triton type states. The amplitudes for the breakup processes can also be related to these amplitudes.

A class of functions having the same analytic properties as the Jost function is introduced. In momentum space these functions can be calculated through solutions of some auxiliary integral equations. The present approach is applicable for both local and nonlocal potentials and suggests an unified calculational scheme for bound states and scattering problems.

The effective two-body NN potential in nuclear matter due to the two-pion-exchange three-body force can be expressed such that it arises due to a change in the pion propagator pertaining to the one-pion-exchange potential. The pion propagator in nuclear matter is examined with the NN correlation taken into account. It is found that the modified propagator can be approximated by ((1 - ..cap alpha..)(q/sup 2/ + ..mu..'/sup 2/))/sup -1/ where ..mu..' = ..mu.. (1 - ..cap alpha..)/sup -1/2/, ..mu.. is the pion mass, and ..cap alpha.. is a positive constant which is roughly proportional to the nuclear matter density. For example, ..cap alpha.. approx. = 0.3 for the normal density of nuclear matter. The modified one-pion-exchange potential in nuclear matter is then obtained by scaling the one-pion-exchange potential in vacuum. It is suggested that other parts of the NN potential, e.g., the two-pion-exchange part, are also modified by the same mechanism, and hence the NN interaction in nuclear matter could be strongly density dependent.

A recently proposed separable expansion for the t -matrix for local potentials is extended to the coupled channel of nucleon-nucleon interactions. The simple method yields a rather convergent separable expansion, with simple form factors, and the requirements of two particle unitarity and time reversal symmetry are maintained for approximations of any rank. The method is illustrated for the simplified Reid soft-core potential in the 3 S 1 - 3 D 1 channel, and the results compared with a recent calculation of Pieper.

Possible alternative interpretations of the Lippmann-Schwinger integral equation for multiparticle (ng2) systems are investigated and are shown to be equivalent if integrals which occur are uniformly convergent, as is reasonable. At real energies E, the derivation of the Lippmann-Schwinger equation from the Schr\"odinger equation involves various surface integrals at infinity in configuration space. It is shown that the values of these surface integrals are related to the values of certain volume integrals at complex energies (E+i\ensuremath{\epsilon}) in the limit \ensuremath{\epsilon}\ensuremath{\rightarrow}0, originally examined by Lippmann. It is further proved that a number of these surface integrals vanish together, a result which---though plausible---previously had to be assumed. The results of this paper confirm previous studies showing that the solutions to the multiparticle Lippmann-Schwinger equation need not be unique. Because of certain convergence difficulties which can occur, the analysis of this paper is not wholly valid for ``three-body'' collisions (defined as collisions involving three independently incident aggregates of the fundamental particles comprising the multiparticle system), or for the even more complicated collisions involving ng3 incident aggregates.

A recently proposed method for t matrix Lippmann-Schwinger-type equations is reexamined and is investigated numerically. The method relies on the introduction of an auxiliary equation containing an arbitrary function, whose kernel is free from the fixed point singularity of the original equation. In the present work we rewrite the final result of the method in such a way that it has all the important features of a related method by Kowalski and Noyes and is simple to use in practice. With special choices of the arbitrary function the present method can be considered as an off-shell extension of methods of Bolsterli, of Kowalski, and of Sasakawa. The method also readily gives a practical way of calculating K matrix elements. Using the iterative solution of the auxiliary equation, the method is tested numerically to compute off-shell t matrix elements for three commonly used nucleon-nucleon potentials for various choices of the arbitrary function.

This paper is mainly concerned with the construction and various properties of a new class of antisymmetric, two-fragment, elastic scattering optical potentials. The construction is based directly on the wave function formulation of $N$-particle collision theories, which are antisymmetrized herein using the method of Adhikari and Gl\"ockle. Those theories which are label transforming are considered first. For such theories, the resulting optical potentials are asymmetric but nevertheless real for energies below the inelastic threshold ${E}_{\mathrm{inel}}$, and are asymmetric and absorptive for $E\ensuremath{\ge}{E}_{\mathrm{inel}}$. This unusual feature of asymmetry is a consequence of working directly with $N$-particle collision theories, almost all of which are expressed in non-Hermitian matrix form, and arguments are presented as to why asymmetry is not a practical problem. Not only are the new class of optical potentials asymmetric, it is also found that exchange effects generally enter them in an extremely simple fashion. These latter two features distinguish the members of the new class from the optical potential developed recently by Goldflam and Kowalski. Examples of label-transforming theories producing such potentials are the extended Faddeev theory of Levin, the precursor form of the Bencze, Redish, Sloan theory, and the wave function component theory of L'Huillier, Redish, and Tandy. Their existence establishes that formalisms other than those based on the Alt, Grassberger, Sandhas transition operator lead to symmetrized optical potentials free of elastic unitarity cuts, albeit potentials which are not Hermitian analytic. In addition to the above theories, a new, symmetrized form of the equations of the channel permuting array theory is developed.

We propose an effective range expansion valid for systems with small scattering length a (|a| → 0) interacting via an attractive short-range and a repulsive Coulomb potential.

With use of a simple version of the optical background representation for the transition amplitude in the case of well nested doorways an alternate statistical theory of multistep compound reactions is suggested, which is valid under very general conditions. Simple connections are established between the present approach and that of Feshbach, Kerman, and Koonin and of Agassi, Weidenmueller, and Mantzouranis.

We remark that resonances in the /sup 12/C(/sup 16/O,/sup 8/Be)/sup 20/Ne reaction are due to the ''exit doorway state,'' that is ready for decay to the exit channel. We find a parameter-free formula that relates the resonance energy and angular momentum of the /sup 12/C+/sup 16/O system to those of the /sup 12/C+/sup 12/C system. If we use the experimental resonance energies of /sup 12/C+/sup 12/C, this formula yields more than a dozen one-to-one correspondences to the resonance energies of /sup 12/C(/sup 16/O,/sup 8/Be)/sup 20/Ne. We also find a parameter-free formula for /sup 12/C(/sup 16/O,/sup 4/He)/sup 24/Mg.

The role of the deuteron asymptotic D to S normalization ratio eta/sup d/ in the calculation of triton observables is emphasized both within a simple model and in an exact numerical calculation. We suggest a new correlation among eta/sup t//eta/sup d/ and E/sub t/ in dynamical three nucleon calculations, where eta/sup t/ is the triton asymptotic D to S ratio and E/sub t/ is the triton binding energy. Studying this correlation we obtained eta/sup t//eta/sup d/ = 1.68 +- 0.04. .AE

A recent work by Rosenberg on cluster states in reaction theory is reexamined and generalized to include energies above the threshold for breakup into four composite fragments. The problem of elastic scattering between two interacting composite fragments is reduced to an equivalent two-particle problem with an effective potential to be determined by extremum principles. For energies above the threshold for breakup into three or four composite fragments effective few-particle potentials are introduced and the problem is reduced to effective three- and four-particle problems. The equivalent three-particle equation contains effective two- and three-particle potentials. The effective potential in the equivalent four-particle equation has two-, three-, and four-body connected parts and a piece which has two independent two-body connected parts. In the equivalent three-particle problem we show how to include the effect of a weak three-body potential perturbatively. In the equivalent four-body problem an approximate simple calculational scheme is given when one neglects the four-particle potential the effect of which is presumably very small.NUCLEAR REACTIONS Scattering theory. Effective few-particle formulation of multiparticle problems. Approximation schemes.

We study the correlation among s-wave low-energy four-nucleon observables by using a simplified dispersion theoretic (N/D) approach. We find that the triton binding energy, its asymptotic normalization parameter, the deuteron exchange left-hand cut, and the nucleon-trinucleon (N-t) scattering length control the low-energy N-t system. The isospin one channel is insensitive to the asymptotic normalization parameter of the triton. The effective range function k cotdelta for the isospin zero spin singlet N-t system has a pole whose position and residue are strongly correlated with the N-t scattering length. In this case the four-nucleon system (in our model) has an excited state whose binding energy and asymptotic normalization parameter are also correlated with the N-t scattering length.

Certain versions of distorted and reduced scattering equations are found to admit continuum spurious solutions apart from the commonly known discrete spurious solutions. The spurious threshold corresponding to continuum spurious solutions may occur at an energy lower than the physical bound state energy and this may make the use of such equations questionable both for bound state and scattering problems. The spurious threshold, however, disappears in approximate calculation using bound state approximation. The origin of these spuriosities lies in a different type of factorization than that found by Newton in the context of the formulation of Weinberg and may shed light on the spurious spectrum of other formulations.

We present an alternative derivation of the necessary set of three-body Lippmann-Schwinger equations, which makes use of the asymptotic form of three-body scattering states in various channels and sheds light on certain possible traps. Thereby making explicit use of the identities for the channel resolvent operators, the homogeneous Lippmann-Schwinger equations are shown to be correct and consistent with the inhomogeneous one, contrary to a recent suggestion by Mukherjee.

After Faddeev and Yakubovskii showed how to write connected few-body equations which are free from discrete spurious solutions various authors have proposed different connected few-body scattering equations. Federbush first pointed out that Weinberg's formulation admits the existence of discrete spurious solutions. In this paper we investigate the possibility and consequence of the existence of spurious solutions in some of the few-body formulations. Contrary to a proof by Hahn, Kouri, and Levin and by Bencze and Tandy the channel coupling array scheme of Kouri, Levin, and Tobocman which is also the starting point of a formulation by Hahn is shown to admit spurious solutions. We can show that the set of six coupled four-body equations proposed independently by Mitra, Gillespie, Sugar, and Panchapakesan, by Rosenberg, by Alessandrini, and by Takahashi and Mishima and the seven coupled four-body equations proposed by Sloan are related by matrix multipliers to basic sets which correspond uniquely to the Schr\"odinger equation. These multipliers are likely to give spurious solutions to these equations. In all these cases spuriosities are shown to have no hazardous consequence if one is interested in studying the scattering problem.NUCLEAR REACTIONS Scattering theory, spurious solutions in three- and four-body equations.

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind gave very good results for Lippmann-- Schwinger equations. The method is here extended to give a very simple and practical method for the three-body equations with separable two-body interactions. Numerical calculations of the three-body amplitudes are carried out at positive energies for two simple models. In spite of the more complicated singularities in the kernel as well as in the solutions of the equation, satisfactory convergence is achieved.

A general formalism is proposed for evaluating the effective interaction between valence nucleons, taking into consideration the low-lying contributions of two-particle two-hole (2p2h), two-particle one-hole (2p1h), three-particle one-hole (3p1h), three-particle two-hole (3p2h), and four-particle two-hole (4p2h) correlations. The formalism consists of solving a hierarchy of multiple scattering type equations with coupling between various correlations and is naturally combined with the Q-box approach for folded diagram theory. The energy dependence of the reaction matrix is included in the formalism. The effect of 4p2h correlations is studied for a simple two-level model for mass 42 by solving the hierarchy of coupled equations numerically by neglecting the coupling to 2p1h and 3p1h configurations. The 4p2h correlation with 4p coupled to isospin 0 has dominant contributions as expected. For a reasonable choice of the energy denominator, the 4p2h contribution to the two-body effective interaction is sizably attractive.