MULTICHANNEL EFFECTIVE-RANGE THEORY WITH LONG-RANGE INTERACTIONS

Effective-range theory, for both single-channel and multichannel scatt ering by systems interacting with shea-range forces at low energies, p rovides a simple representation of the energy dependence of scattering matrix elements near reaction thresholds in conformity with the Wigne r threshold law. A modification of effective-range theory for single-c hannel electron-atom scattering, required to account for long-range po larization forces, was developed some years ago. A multichannel extens ion of that theory is presented here, allowing for a superposition of asymptotic power-law potentials, with asymptotic wave functions expres sed as sums of Bessel functions. Threshold branch-point singularities are extracted explicitly, leaving modified reaction-matrix elements th at vary smoothly with energy. For tar et states that are not spherical ly symmetric the asymptotic wave functions account for inverse-cube la w interactions that couple the various channels. The theory provides a n unambiguous prescription for analytic continuation of scattering par ameters from just above to just below a reaction threshold, a feature that may be useful, for example, in studying resonant cross sections b ased on above-threshold calculations. The present work represents an e xtension, to a wider class of scattering systems and long-range intera ctions, of an earlier analysis of threshold behavior [M. Gailitis and R. Damburg, Proc. Phys. Sec. London 82, 192 (1963)] applicable to elec tron-hydrogen scattering in the field of the inverse-square potential arising from the linear Stark effect.