Surface-integral formulation of scattering theory

Abstract

a b s t r a c tWe formulate scattering theory in the framework of a surface-integral approach utilizing analytically known asymptotic forms of the two-body and three-body scattering wavefunctions. This formulation is valid for both short-range and long-range Coulombic interactions. New general definitions for the potential scattering amplitude are presented. For the Coulombic potentials, the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the Coulomb three-body breakup amplitude are derived. This resolves the problem of the inability of the conventional scattering theory to define the post form of the breakup amplitude for charged particles. The new definitions can be written as surfaceintegrals convenient for practical calculations. The surface-integral representations are extended to amplitudes of direct and rearrangement scattering processes taking place in an arbitrary three-body system. General definitions for the wave operators are given that unify the currently used channel-dependent definitions.

Surface-Integral Approach to the Coulomb Few-Body Scattering Problem

“…(12)). The usage of the replacement (35) is discussed in [12]. It is not difficult to see that the results given above are consistent with conventional potential scattering theory for short-range interactions.…”

“…(32) represents the logical fact that the unscattered incident wave is coming from infinity and should be taken in a form valid at asymptotically large distances (see [12] for more discussion). With these prerequisites repeating the algebra of the previous subsection we get…”

Coulomb-distorted plane wave: Partial wave expansion and asymptotic forms

“…Mathematical interpretation of this result was presented in Section 5 of Ref. 6. The asymptotic expansion can be interpreted in distributional sense and the properties of the considered test function space plays an important role.…”

“…However, the surface integral formalism, as explained in Ref. 6, depends only on the validity of (12).…”

“…5 and 6 we will call the asymptotic state of Mulherin and Zinnes 3 as Coulomb-distorted plane wave (CDPW). The recently developed surface integral formalism of quantum scattering theory [5][6][7] is valid not only for two-body scattering but it can be extended to scattering of three charged particles. Another advantage of this formalism is that it handles the short and long range interactions on an equal footing.…”