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  • 1
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 2648-2658 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The radial equation (or set of equations) derived in scattering theory is analyzed by means of Titchmarsh–Weyl theory for singular second-order differential equations. In particular we have focused on the spectral density concept and the corresponding relation to the scattering cross section. The method of complex deformations is brought in as a necessary ingredient in the evaluation of the underlying pole strings, which together with the background build up the actual dispersion relation data. The analysis is supported by numerical applications to a centrifugal family of simple potentials.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 2629-2639 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Titchmarsh–Weyl theory is applied to the Schrödinger equation in the case when the asymptotic form of the solution is not known. It is assumed that the potential belongs to the Weyl's limit-point classification. A rigorous analytical continuation of the Green's function, obtained from the solution regular at the origin and the square integrable Weyl's solution (regular at infinity), to the "unphysical'' Riemann energy sheet is carried out. It is demonstrated how the Green's function can be uniquely constructed from the Titchmarsh–Weyl m-function and its Nevanlinna representation. The behavior of the m-function in the neighborhood of poles is investigated. The m-function is decomposed in a, so called, generalized real part (Reg) and a generalized imaginary part (Img). Reg(m) is found to have a significant argument change upon pole passages. Img(m) is found to be a generalized spectral density. From the generalized spectral density, a spectral resolution of the differential operator and its resolvent is derived. In the expansion contributions are obtained from bound states, resonance states (Gamow states), and the "deformed continuum'' given by the generalized spectral density.The present expansion theorem is applicable to the general partial differential operator via a decomposition into partial waves. The numerical procedure involves all quantum numbers l and m, but for convenience, and with the inverse problem in mind, this study is focused on the case when the rotational quantum number equals zero. The theory is tested numerically and analyzed for an analytic model potential exhibiting a barrier and decreasing exponentially at infinity. The potential is Weyl's limit point at infinity and allows for an analytical continuation into a sector in the complex plane. An attractive feature of the generalized spectral density of the present potential is that the poles close to the real axis seem to exhaust or deflate the above-mentioned density inside the pole string. Outside this string the density rapidly approaches that of a free particle. This information is used to derive an approximate representation of the m-function in terms of poles and residues as well as free-particle background. In order to display the features mentioned above, the present study is accompanied with several plots of analytically continued quantities related to the Green's function.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Description / Table of Contents: La méthode AMO, modifiée à l'aide d'un “paramètre” d'échelle a été appliquée à l'état 1sσ2pσ1∑u+ de la molécule d'hydrogène. On emploie une méthode pour étendre le domaine du “paramètre de mélange” dans le plan complexe. Tous les paramètres employés ont été complètement variés. Pour les grandes séparations internucléaires on obtient une amélioration considérable de l'énergie totale. Les résultats obtenus ont été comparés aux ceux de Kolos et Wolniewicz en fonction de la distance internucléaire R. Pour R = 12 u.a. la différence entre les deux énergies n'est que 18% de la différence correspondante pour R = 2.43 u.a. Pour la distance internucléaire à l'équilibre on trouve 2.140 u.a. ce qui n'est pas en très bon accord avec la valeur 2.429 u.a., obtenue par les deux autres auteurs.Dans la limite des atomes séparés l'état en question ne donne pas H+ + H-, bien que le caractère ionique de la fonction d'onde soit dominant dans la région 3 ≦ R ≦ 8 u.a.
    Abstract: Es wurde eine Version der AMO-Methode mit einen Koordinatenstreckungsparameter auf den 1sσ2pσ1∑u+ Zustand des Wasserstoffmoleküls angewendet. Alle eingeführten Parameter wurden vollstündig variiert. Für grosse Kernabstände, R, wurde dadurch eine ansehnliche Verbesserung der berechneten Energiewerte erhalten. Die hier berechnete Potentialkurve wurde mit der von Kolos und Wolniewicz berechneten verglichen. Für r = 12 a.E. ist die Energie-differenz nur 18% von der für R = 2.43 a.E. erhaltenen. Für den Gleichgewichtskernabstand gab unsere Berechnung 2.140 a.E., was in schlechter Übereinstimmung mit dem von Kolos und Wolniewicz erhaltenen Abstand, 2.429 a.E. steht.In der Grenze getrennter Atome wurde der hier behandelte Zustand nicht in H+ + H- dissoziiert, obgleich der Ionencharacter der Wellenfunktion im Intervall 3 ≦ R ≦ 8 a.E. dominierend sei.
    Notes: A scaled version of the AMO method is applied to the 1sσ2pσ 1∑u+ state of the hydrogen molecule. A method to extend the domain of the mixing parameter λ to the whole complex plane is described and applied in the present calculation. All the parameters introduced have been varied completely. A considerable improvement in the computed energy values is found for large internuclear separations R. The comparison between our potential energy curve and the accurate curve calcualted by Kolos and Wolniewicz is studied and, for example, for R = 12 a.u. the energy difference is only 18% of that for R = 2.43 a.u. The equilibrium separation is found to be 2.140 a.u. in poor agreement with 2.429 a.u. obtained by the previously mentioned authors.In the separated atom limit, the state under consideration does not dissociate into H+ + H-, although the ionic character of the wave function is dominating in the region 3 ≦ R ≦ 8 a.u. The connections with earlier calculations and methods, especially the scaled version of the MO-LCAO approximation, are also pointed out and discussed.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Description / Table of Contents: On discute une application à la molécule d'hydrogène d'une version de la méthode AMO avec un facteur de “scaling”. On donne de l'information additionelle sur des calculs antérieurs et on discute surtout le comportement asymptotique.
    Abstract: Es wird eine Anwendung einer Version der AMO-Methode mit Koordinatenstreckungsfaktor an das Wasserstoffmolekül diskutiert. Zusätzliche Information über frühere Berechnungen wird gegeben. Vor allem wird das asymptotische Verhalten untersucht.
    Notes: The scaled version of the AMO method as applied to the hydrogen molecule is discussed. Additional information about previous calculations on H2 with similar approximations together with computations on excited states (states corresponding to the next lowest root of the secular equation) are reported. The asymptotic behaviour is particularly investigated.
    Additional Material: 3 Tab.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 3 (1969), S. 1069-1070 
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 9 (1975), S. 103-117 
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Weyl's eigenvalue theory for ordinary second-order differential equations is discussed for the case of a continuous spectrum. It is demonstrated that the spectral density function obtained from a suitably averaged Green's function, equal to the Weinstein function, can be directly related to the Weyl-Titchmarsh m-function. The explicit connections with scattering theory are derived and it is found that the Weyl and Jost solutions are proportional; the proportionality factor being the reciprocal value of the latter at the origin. The physical interpretation of the complex poles of the spectral density is discussed in relation to Gamow's exponentially growing functions. The advantage of using a formulation that allows for a “perturbation” of boundary conditions is pointed out.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 23 (1983), S. 865-874 
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: The Titchmarsh-Weyl theory for a singular second-order differential equation is presented. The equations, however, are arranged such that the theory is immediately applicable to the multichannel formulation. The mathematical basis of the Titchmarsh-Weyl theory (TWT) is outlined and the Titchmarsh-Weyl m function is presented. Relations to the associated Green's function and the spectral function are given. It is shown how some quantities common in scattering theory can be expressed in terms of the TWT. The method of complex rotation is then applied to the TWT to create a complex rotated analog of the Titchmarsh-Weyl theory. This is then extended as exterior complex scaling is introduced. An outline of a proof for the method of exterior complex scaling is presented and some numerical results of the complete theory are given.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 22 (1982), S. 445-457 
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Weyl's theory for a set of N-coupled singular second-order differential equations is analyzed in relation to S-matrix theory and a dilated version is presented. Applications of this theory to two single channel scattering model problems and a two-channel model problem are given. Some implications of the present theory are discussed.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 23 (1983), S. 91-98 
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: The self-consistent field (SCF) equations for many-electron systems, suitable within the complex-coordinate method, are derived. The formulation is based on a general bivariational theorem for non-Hermitian operators, with an emphasis on the analytic structure invoked by the complex dilation of the total Hamiltonian. The dilation structure of the resulting SCF equations is stressed and the concomitant analytical properties are discussed. The solutions are classified with respect to these properties, and interpreted in terms of a general form of the symmetry dilemma. The role of the dilated SCF equations for resonance calculations is discussed.
    Type of Medium: Electronic Resource
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