The widely used continuum discretized coupled-channels (CDCC) method for approximate calculations of three-body systems is discussed as a truncation of a Faddeev formulation in angular momentum space. A set of coupled equations is presented for converting a CDCC solution into a full solution of the original Schrodinger equation. A practical iterative procedure for solving the equations is outlined, based on the "distorted Faddeev equations" of Birse and Redish.PACS numbers: 24.10.-i, 03.80.+r, 1 l.SO.Jy, 34.10.+X Success of the method of continuum discretized coupled channels (CDCC) in describing direct nuclear reactions involving breakup of colliding nuclei has motivated more general inquiries into its foundations as a numerical procedure for solving many-body scattering problems. When applied to a three-body system this method solves the Schrodinger equationin a restricted model space, as described at length in recent reviews. ^' Here the notation refers to the following explicit, illustrative problem: A neutron n and a proton /?, with coordinates Tp and r", move in the vicinity of an infinitely massive nucleus A, located at the origin. The kinetic-energy operator for the nucleons is K, the interaction potential between then is K, and the interaction potentials between A and the individual nucleons are Up and Un-For simplicity we assume that K is a shortranged central potential. We also assume that incident waves are present only in the deuteron channel. Spins are ignored. Recent criticisms of the CDCC method consider the possible sensitivity of the results to the choice of model space.^'"^ Discussions of CDCC foundations^'^ emphasize the importance of complex optical potentials Up.Un as a justification for the central approximations. We recall^ in the present paper that the CDCC method can be derived as a simple truncation of an orderly Faddeev-type formulation. Hence the method is recognized as a practical, physically motivated approximation procedure within the general three-body theory, and this eliminates much of the vagueness of the recent analyses of CDCC. We go on to propose improved procedures and new applications for this approach to three-body analyses.The CDCC model space is defined primarily by the unusual projection operator P that only selects low angular momenta / associated with the neutron-proton relative coordinate r =rp -T;,, up to a maximum /^. We will see that the basic properties of the model-space theory are obtained under any finite choice of Im-Often the parameter Im is taken to be only 0 or 2. We also define \-P-Q, Further properties of the model space appear later when discretization of the P-space continuum is introduced; the resulting finite number of "discretized continuum channels" comprise the standard model space.Our principal point is that the partial wave functions P\l/ and Qxj/ not only are orthogonal, but they also meet the essential mathematical requirements for Faddeev components. Namely, the asymptotic two-body channels in distinct partitions of the system are located un...
An eigenfunction of an attractive nonlocal single-particle potential is always smaller inside the region of the potential than outside; the converse occurs for repulsive potentials. This is the Perey effect. In the present article explicit formulas for the effect are derived for the case of motion in one dimension, and interpretive discussions of the effect are given. The derivation does not employ series expansions. It is argued that the effect can be understood in terms of the fundamental many-body theory from which the single-particle potential has (in principle) been derived. Some of the wave function lies in the channels which have been eliminated in the course of that derivation.
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