Unitarity constraints in modified effective-range theory

Abstract

The effect of a long-range polarization force on the low-energy scattering of an electron by a neutral atom may be accounted for by adopting a form of distorted-wave theory in which the phase shift is represented as the sum of a polarization phase and a relative phase. From this latter phase an effective-range function is formed having a smooth energy dependence in the neighborhood of threshold. Multichannel extensions of this approach have been described. For highly polarizable targets, and for scattering energies sufficiently removed from threshold, the polarization phase takes on complex values. The relative phase, and hence the effective-range parameters, must then be complex as well since unitarity requires that the physical phase shift be real. For scattering by short-range potentials unitarity is preserved by formulating approximation procedures in terms of a Hermitian reaction matrix. This standard approach is generalized here in the context of effective-range theory involving strong polarization forces. The hermiticity condition on the reaction matrix is replaced by a more general requirement relating its Hermitian and anti-Hermitian components. Methods for imposing unitarity constraints in approximation procedures are described and illustrated, for resonant and nonresonant systems, in numerical examples.