Analytic properties of the S matrix for a simple model of fixed-nuclei electron-polar-molecule scattering

The square-well potential augmented by a spherically symmetric r ^-2 potential tail is reconsidered as a simple model of the electron-polar-molecule interaction. The complete solution of the s-wave scattering problem is presented with particular emphasis on the analytic properties of the scattering matrix element in the complex momentum plane. Resonances, bound states and virtual states and their trajectories as a function of the strength parameter of the short-range potential are investigated. The recent dispute about the existence or non-existence of virtual states in models of this type is settled. Numerical calculations of the low-energy phaseshift and cross section for various strengths of the square well and the r^-2 tail are presented. These data should prove useful in interpreting, in simple terms, ab initio calculated eigenphases for electron-polar-molecule scattering at low energy.