S ‐matrix expansions in view of complex dilation theories

2005

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ABSTRACT:A two-channel problem is considered within a method based on firstorder differential equations that are equivalent to the corresponding Schrödinger equation but more convenient for dealing with resonant phenomena. Using these equations, it is possible to calculate directly the Jost matrix for practically any complex value of the energy. The spectral points (bound and resonant states) can therefore be located in a rigorous way, namely, as zeros of the Jost matrix determinant. When calculating the Jost matrix, the differential equations are solved and thus, at the same time, the wave function is obtained with the correct asymptotic behavior analytically embedded in the solution. The method offers a very accurate way of calculating not only total widths of resonances but their partial widths as well. For each pole of the S-matrix, its residue can be calculated rather accurately, which makes it possible to obtain the Mittag-Leffler-type expansion of the Smatrix as a sum of the singular terms (representing the resonances) and the background term (contour integral). As an example, the two-channel model by Noro and Taylor is considered. It is demonstrated how the contributions of individual resonance poles to the scattering cross section can be analyzed using the Mittag-Leffler expansion and the Argand plot technique. This example shows that even poles situated far away from the physical real axis may make significant contributions to the cross section.

Section: Introductionmentioning

confidence: 71%

“…Based on the Titchmarsh–Weyl theory, a method for decomposing the scattering information into the resonance and background contributions was developed 41. One of us (N. E.) analyzed the Mittag–Leffler method in an earlier study, to understand its computational properties 19–21. In Ref.…”

Section: Introductionmentioning

confidence: 99%

Analyzing the contribution of individual resonance poles of the S‐matrix to two‐channel scattering

Rakityansky

2005

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“…(19) from zero to η r and from η r to ∞ along the complex path with the argument θ gives W [ f ( k , η r ), φ( k ′, η r )]. Differentiation with respect to k ′ and with k ′ = k j leads to Hence, residues of the partial wave S ‐matrix inspired by 2 and as presented in 24 can be reduced to …”

Section: Theorymentioning

confidence: 99%

Role of resonances in building collision cross‐sections: Mittag‐Leffler‐based study

Alferova

2003

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.

“…Based on the Titchmarsh–Weyl theory, a method was developed 20 for decomposing the scattering information into resonance and background (free‐particle) contributions. This has led us to try to find methods that can be used to determine the role of resonances in a collision cross section 21–24.…”

Section: Introductionmentioning

confidence: 99%

Role of resonances in building cross sections: Comparison between the Mittag–Leffler and the T‐matrix Green function expansion approaches

Shilyaeva

Yarevsky

2006

Peaks in collision cross sections are often interpreted as resonances.The complex dilation method, as well as other methods relying on analytic continuation of the scattering formalism, can be used to clarify whether these structures are true resonances in the sense that they are poles of the S-matrix and the associated Green function. The performance of the Mittag-Leffler expansion and T-matrix Green function expansion methods are formally and computationally compared. The two methods are applied to two model potentials. Eigenenergies, s-wave residues, and cross sections are computed with both methods. The resonance contributions to the cross sections are further analyzed by removing the residue contributions from the Mittag-Leffler and Green function expansion sums, respectively. It is suggested that the contribution of a resonance to a cross section should be defined through its S-matrix residue.