Positive energy Weinberg states for the solution of scattering problems

Positive energy Weinberg states are defined and numerically calculated in the presence of a general complex Woods-Saxon potential. The numerical procedure is checked for the limit of a square well potential for which the Weinberg states and the corresponding eigenvalues are known. A finite number M of these (auxiliary) positive energy Weinberg states are then used as a set of basis functions in order to provide a separable approximation of rank M, VM, to a potential V, and also to the scattering matrix element S which obtains as a result of the presence of V,SM. Both VM and SM are obtained by means of algebraic manipulations which involve the matrix elements of V calculated in terms of the auxiliary positive energy Weinberg states. Next, expressions are derived which enable one to iteratively correct for the error in V-VM. These expressions are a modified version of the quasiparticle method of Weinberg. The convergence of SM to S, as well as the first order iteration of the error in SM, is examined as a function of M for a numerical example which uses a complex Woods-Saxon potential for V and assumes zero angular momentum. With $M=5$ and one iteration an error of less than 10% in S is achieved; for $M=8$ the error is less than 1%. The method is expected to be useful for the solution of large systems of coupled equations by matrix techniques or when a part of the potential is nonlocal.NUCLEAR REACTIONS Scattering theory, expansions on a basis set of positive energy Weinberg states, removal of truncation error by the quasiparticle method. Normal mode effective scattering channels.