Siegert pseudostate formulation of scattering theory: Two-channel case

Abstract

Siegert pseudostates (SPS) are a finite basis representation of Siegert states (SS) for finite-range potentials. This paper presents a generalization of the SPS formulation of scattering theory, originally developed by Tolstikhin, Ostrovsky, and Nakamura [Phys. Rev. A 58, 2077 (1998)] for s-wave scattering in the one-channel case, to s-wave scattering in the two-channel case. This includes the investigation of the properties of orthogonality and completeness of two-channel SPS and the derivation of the SPS expansions for the two-channel Green function, wave function, and scattering matrix. Similar to the one-channel case, two types of expansions for the scattering matrix are obtained: one has a form of a sum and requires the knowledge of both the SPS eigenvalues and eigenfunctions, while the other has a form of a product and involves the eigenvalues only. As the size of the basis tends to infinity, the product formulas obtained here in terms of SPS coincide with those given by Le Couteur [Proc. R. Soc. London, Ser. A 256, 115 (1960)] in terms of SS; all the other relations, as far as we know, have no counterparts in the literature. Partial widths of resonances in the case when both channels are open for decay are identified in terms of SPS---a feature that is absent in the one-channel case. The results are illustrated by numerical calculations for two model potentials.