Siegert pseudostate formulation of scattering theory: One-channel case

Abstract

Siegert pseudostates (SPSs) are defined as a finite basis representation of the outgoing wave solutions to the radial Schr\"odinger equation for cutoff potentials and the problem of their calculation is reduced to standard linear algebra easily implementable on computers. For a sufficiently large basis and the cutoff radius, the set of SPSs includes bound, weakly antibound, and narrow complex-energy resonance states of the system, i.e., all the physically meaningful states observable individually. Moreover, the set is shown to possess certain orthogonality and completeness properties that qualify it as a discrete basis suitable for expanding the continuum. We rederive many results of the theory of Siegert states in terms of SPSs and obtain some (to our knowledge) previously unknown relations. This not only makes the results practically applicable, but also sheds a new light on their mathematical nature. In particular, we show how the Mittag-Leffler expansions for the outgoing wave Green's function and the scattering matrix can be obtained on the basis of very simple algebraic relations, without assuming them to be meromorphic functions. Explicit construction of these two fundamental objects completes the SPS formulation of scattering theory for the one-channel case. The computational efficiency of this approach is illustrated by a number of numerical examples.