Calculation of Substituted Fredholm Determinants Using Complex Basis Functions

Using basis functions with complex coordinates, it is possible to construct discrete approximations to the Fredholm determinant directly at real energies using only square-integrable functions. For the case of purely elastic scattering, the procedure is equivalent to an analytic continuation of the coordinate dependence of the Hamiltonian. It is shown that improved convergence is obtained by an application of the "dispersion-correction" method. The method is generalized to allow calculation of the "substituted" Fredholm determinants needed to construct the $S$ matrix for many-channel potential-scattering problems. This generalization is not equivalent to a simple continuation of the coordinate dependence of the many-channel Hamiltonian. Results of calculations on several model problems are presented.