Long-range behavior of the optical potential for the elastic scattering of charged composite particles

The asymptotic behavior of the optical potential, describing elastic scattering of a a charged particle {α} off a bound state of two charged, or one charged and one neutral, particles at small momentum transfer Δ_α or, equivalently, at large intercluster distance ρ_α, is investigated within the framework of the exact three-body theory. For the three-charged-particle Green function that occurs in the exact expression for the optical potential, a recently derived expression, which is appropriate for the asymptotic region under consideration, is used. We find that for arbitrary values of the energy parameter, the nonstatic part of the optical potential behaves for Δ_α{→}0 as C₁{{{Δ}}}_{{{α}}}$+o(${{{Δ}}}_{{{α}}}$). From this we derive for the Fourier transform of its on-shell restriction for ${{{ρ}}}_{{{α}}}→∞ the behavior -a/2{{{ρ}}}_{{{α}}}⁴$+o(1/${{{ρ}}}ₐ⁴$), i.e., dipole or quadrupole terms do not occur in the coordinate-space asymptotics. This result corroborates the standard one, which is obtained by perturbative methods. The general, energy-dependent expression for the dynamic polarizability C₁ is derived; on the energy shell it reduces to the conventional polarizability a which is independent of the energy. We emphasize that the present derivation is nonperturbative, i.e., it does not make use of adiabatic or similar approximations, and is valid for energies below as well as above the three-body dissociation threshold.