ABSTRACT:The method of complex dilation is used to define the partial wave Smatrix in the sector of the fourth quadrant of the complex energy plane. Two ways of obtaining the expansion coefficients-the partial wave S-matrix residues-are studied. The Mittag-Leffler decomposition of the partial wave S-matrix as a sum of residue terms and an integral contribution is used to define the contributions of a number of the partial wave S-matrix poles, related to a 1-D potential, to the corresponding S-wave cross-section. The obtained expansion demonstrates a way of describing the contribution from a single pole to the partial wave S-matrix and thereby to various types of cross-sections. Our model study shows how peaks in a cross-section not only can be attributed to so-called isolated resonances but also to a set of overlapping barrier-type resonances.

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ABSTRACT:In this article, we use the complex dilation formalism to compute Smatrix 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 Ᏺ ϩ (k) and Ᏺ Ϫ (k) as well as the derivative dᏲ/dk numerically at the eigenvalue k . 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.

ABSTRACT:A method using a superposition of a finite but varied number of ␦-potentials has been applied to treat the integral Schrö dinger equation with three different potentials. Positions of scattering resonances are numerically determined as complex poles of the partial-wave S-matrix. The method, in principle, makes it possible to obtain several resonances for short-range potentials like a square well (or barrier). The numerical method is elaborated to treat integral equations and is applicable to solve problems for which no differential equations can be derived.

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