  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 1/r^2 singular short-range potentials
but confined ourselves to the sub-critical coupling regime where the coupling
parameter strength of the 1/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 and by coping with slower than usual
convergence. In a follow-up study, we intend to apply the method to obtain
scattering information for realistic potential models.
