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 $M$, $V_M$, to a potential $V$, and also to the scattering matrix element $S$ which obtains as a result of the presence of $V$,$S_M$. Both $V_M$ and $S_M$ 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 $V-V_M$. These expressions are a modified version of the quasiparticle method of Weinberg. The convergence of $S_M$ to $S$, as well as the first order iteration of the error in $S_M$, 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 $M=5\mathrm{}$ and one iteration an error of less than 10% in $S$ is achieved; for $M=8\mathrm{}$ 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.