J-matrix method of scattering for inverse-square singular potentials with supercritical coupling I. No regularization

Abstract

The J-matrix method of scattering was developed to handle regular short-range potentials with applications in atomic, nuclear and molecular physics. Its accuracy, stability, and convergence properties compare favorably with other successful scattering methods. It is an algebraic method, which is built on the utilization of orthogonal polynomials that satisfy three-term recursion relations and on the manipulation of tridiagonal matrices. Recently, we extended the method to the treatment of r−2 singular short-range potentials but confined ourselves to the sub-critical coupling regime where the coupling parameter strength of the r−2 singularity is greater than −1/8. In this work, we expand our study to include the supercritical coupling in which the coupling parameter strength is less than −1/8. However, to accomplish that we had to extend the formulation of the method to objects that satisfy five-term recursion relations and matrices that are penta-diagonal. It is remarkable that we could develop the theory without regularization or self-adjoint extension, which are normally needed in the treatment of such highly singular potentials. Nonetheless, we had to pay the price by extending the formulation of the method into this larger representation space and by coping with slower than usual convergence. In the paper that follows this one, we present an alternative version of the theory where we perform regularization in an attempt to avoid the slow convergence and restore the conventional tridiagonal representation.