Three-potential formalism for the three-body scattering problem with attractive Coulomb interactions

Abstract

A three-body scattering process in the presence of Coulomb interaction can be decomposed formally into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The corresponding integral equations are coupled Lippmann-Schwinger and Faddeev-Merkuriev integral equations. We solve them by applying the Coulomb-Sturmian separable expansion method. We present elastic scattering and reaction cross sections of the e + + H system both below and above the H(n = 2) threshold. We found excellent agreements with previous calculations in most cases. PACS number(s): 34.10.+x, 34.85.+x, 21.45.+v, 03.65.Nk, 02.30.Rz, 02.60.Nm The three-body Coulomb scattering problem is one of the most challenging long-standing problems of nonrelativistic quantum mechanics. The source of the difficulties is related to the long-range character of the Coulomb potential. In the standard scattering theory it is supposed that the particles move freely asymptotically. That is not the case if Coulombic interactions are involved. As a result the fundamental equations of the three-body problems, the Faddeev-equations, become illbehaved if they are applied for Coulomb potentials in a straightforward manner.The first, and formally exact, approach was proposed by Noble [1]. His formulation was designed for solving the nuclear three-body Coulomb problem, where all Coulomb interactions are repulsive. The interactions were split into short-range and long-range Coulomb-like parts and the long-range parts were formally included in the "free" Green's operator. Therefore the corresponding Faddeev-Noble equations become mathematically wellbehaved and in the absence of Coulomb interaction they fall back to the standard equations. However, the associated Green's operator is not known. This formalism, as presented at that time, was not suitable for practical calculations.In Noble's approach the separation of the Coulomb-like potential into short-range and long-range parts were carried out in the two-body configuration space. Merkuriev extended the idea of Noble by performing the splitting in the three-body configuration space. This was a crucial development since it made possible to treat attractive Coulomb interactions on an equal footing as repulsive ones. This theory has been developed using integral equations with connected (compact) kernels and transformed into configuration-space differential equations with asymptotic boundary conditions [2]. In practical calculations, so far only the latter version of the theory has been considered. The primary reason is that the more complicated structure of the Green's operators in the kernels of the Faddeev-Merkuriev integral equations has not yet allowed any direct solution. However, use of integral equations is a very appealing approach since no boundary conditions are required.Recently, one of us has developed a novel method for treating the three-body problem with repulsive Coulomb interactions in three-potential picture [3]. In this approach a three-body Coulomb scattering process can ...

Resonant-state solution of the Faddeev-Merkuriev integral equations for three-body systems with Coulomb potentials

“…where the corresponding CS matrix elements of the twobody Green's operators in the integrand are known analytically for all complex energies (see [3] and references therein), and thus the convolution integral can be performed also in practice.…”

Faddeev–Merkuriev equations for resonances in three-body Coulombic systems

“…where the corresponding CS matrix elements of the two-body Green's operators in the integrand are known analytically for all complex energies (see [5] and references therein). It is also evident that all the thresholds, corresponding to the poles of g x 1 , are at the right location and therefore this method is especially suited to study near-threshold resonances.…”

Three-body breakup within the fully discretized Faddeev equations

“…In the spin-singlet channel there are no N N bound states, so that all functions |z α k in Eq. (18) are approximated by scattering WP's. It is important to note that as a by-product of our diagonalization procedure one gets simultaneously the discrete representation for N N partial phase shifts δ s (ε s * k ) for all pseudo-states energies (i.e.…”