Weyl's theory and the method of complex rotation

Rittby, Magnus; Elander, Nils; Brändas, Erkki

doi:10.1080/00268978200100431

Cited by 65 publications

(62 citation statements)

References 32 publications

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“…Therefore, formula (37) becomes important for numerical methods. One of us (N. E.) has elsewhere [17,25,26] used the differential Schrö dinger equation to study Jost functions. The idea of the present study is to solve the integral Volterra equation (34) for ⌽ l ( p; z).…”

Section: Numerical Treatmentmentioning

confidence: 99%

Integral equations and complex resonance energies for analytical potentials

Kapshaǐ

Alferova

Elander

2006

Int J of Quantum Chemistry

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It is shown that the Volterra integral equation in combination withcomplex scaling gives a formalism capable to solve Schrö dinger-type problems concerning bound states and resonances. The regular solution of the Schrö dinger equation presented at the Volterra integral equation allows us to define the explicit form of the Jost function analytically continued into the lower half complex momentum plane. The resulting formalism is used to develop a numerical method for finding resonances defined as zeros of the Jost function. The numerical method is tested on several analytical potentials; it gives good results for arbitrary orbital momentum l.

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Section: Numerical Treatmentmentioning

confidence: 99%

Integral equations and complex resonance energies for analytical potentials

Kapshaǐ

Alferova

Elander

2006

Int J of Quantum Chemistry

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“…In this article, by combining the analysis of Rittby et al [18], Krylstedt et al [19], and a generalized bound-state technique [1] we will show that the residues of the partial wave S-matrix can be calculated by integrating the corresponding complex resonant eigenfunctions.…”

Section: Mittag-leffler Expansion For the Partial Wave S-matrixmentioning

confidence: 98%

“…In Refs. [18,19], Elander and collaborators have shown that resonance eigenvalues and eigenfunctions can be used to compute the partial wave S-matrix through a Mittag-Leffler expansion [27] …”

Section: Mittag-leffler Expansion For the Partial Wave S-matrixmentioning

confidence: 99%

“…Numerical techniques based on spline methods [20,21], finite element techniques [22][23][24][25], or global basis set techniques cannot in a simple manner be applicable to the techniques used by Rittby et al [18] and Krylstedt et al [19].…”

Section: Introductionmentioning

confidence: 99%

“…To be specific, the methods used in Refs. [18,19] allow direct numerical computation of S-matrix residues as numerical derivatives of the corresponding Jost function.…”

Section: Introductionmentioning

confidence: 99%

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S‐matrix expansions in view of complex dilation theories

Alferova

Elander

2002

Int J of Quantum Chemistry

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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.

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Time‐Dependent Quantum‐Mechanical Approaches to the Continuous Spectrum: Scattering Resonances in a Finite Box

Hammerich

Muga

Kosloff

1989

Israel Journal of Chemistry

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Several novel aspects of scattering resonances are studied. An expression, valid for a finite box, relating the continuum phase shift with the energy shift and unperturbed level separation is proposed and applied to obtain the resonance parameters. The effect of the resonance on propagating a wavepacket in imaginary time is studied. It is observed that the resonance strongly affects the cumulants ofthe energy distribution. In particular, a local minimum of the first derivative of the energy with respect to time (proportional to the second cumulant) serves to estimate the resonance energy and wavefunction. Once the estimate is known, the autocorrelation function is used to evaluate the resonance width. Alternatively, a new iterative approach is developed that is capable of selectively yielding an arbitrary band of energy eigenvalues and eigenfunctions on a grid. This method is applied to give those energy levels that are of interest for the discrete computation of the resonant phase shift, i.e., those close to resonance. Exact (analytical) and approximate results are in good agreement for a particular separable potential model in one dimension. These methods can be extended to realistic potentials in higher dimensions.

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Weyl's theory and the method of complex rotation

Cited by 65 publications

References 32 publications

Integral equations and complex resonance energies for analytical potentials

Integral equations and complex resonance energies for analytical potentials

S‐matrix expansions in view of complex dilation theories

Time‐Dependent Quantum‐Mechanical Approaches to the Continuous Spectrum: Scattering Resonances in a Finite Box

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