Multichannel effective-range theory with long-range interactions

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Abstract
Effective-range theory, for both single-channel and multichannel scattering by systems interacting with short-range forces at low energies, provides a simple representation of the energy dependence of scattering matrix elements near reaction thresholds in conformity with the Wigner threshold law. A modification of effective-range theory for single-channel electron-atom scattering, required to account for long-range polarization forces, was developed some years ago. A multichannel extension of that theory is presented here, allowing for a superposition of asymptotic power-law potentials, with asymptotic wave functions expressed as sums of Bessel functions. Threshold branch-point singularities are extracted explicitly, leaving modified reaction-matrix elements that vary smoothly with energy. For target states that are not spherically symmetric the asymptotic wave functions account for inverse-cube law interactions that couple the various channels. The theory provides an unambiguous prescription for analytic continuation of scattering parameters from just above to just below a reaction threshold, a feature that may be useful, for example, in studying resonant cross sections based on above-threshold calculations. The present work represents an extension, to a wider class of scattering systems and long-range interactions, of an earlier analysis of threshold behavior [M. Gailitis and R. Damburg, Proc. Phys. Soc. London 82, 192 (1963)] applicable to electron-hydrogen scattering in the field of the inverse-square potential arising from the linear Stark effect.