S-matrix expansions in view of complex dilation theories

Abstract

In this article, we use the complex dilation formalism to compute S-matrix residues for a number of complex Schrödinger-type eigenvalues in a limited spectral region. These residues are then used as expansion parameters for the corresponding S-matrix. The aim of this work is to find numerically stable algorithms that can be applied to problems based on realistic only numerically represented potentials. Two formalisms are described. In the first, we compute the Jost functions ℱ+(k) and ℱ-(k) as well as the derivative dℱ/dk numerically at the eigenvalue kv. Using the second method, we generalize a formula described by Newton [(Scattering Theory of Waves and Particles; McGraw-Hill; New York, 1966) for bound states to resonant eigenvalues. Here, the S-matrix residue is obtained by integrating the resonant eigenfunction in the complex coordinate space. Our formalism is tested on an exactly solvable problem (Bargmann, V. Phys Rev 1949, 75, 301-303) and a classic complex dilation potential (Bain, R.A. et al. J Phys B Atom Mol Phys 1974, 7, 2189-2202). © 2002 Wiley Periodicals, Inc. Int. J. Quantum Chem. 90.