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  • 1
    Electronic Resource
    Electronic Resource
    Moment problem formulation of the simplified ideal magnetohydrodynamics ballooning equation (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 717-720 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A fundamentally new method for determining the eigenvalues of linear differential operators is presented. The method involves the application of moment analysis and affords a fast and precise numerical algorithm for eigenvalue computation, particularly in the intermediate and strong coupling regimes. The most remarkable feature of this approach is that it provides exponentially converging lower and upper bounds to the eigenvalues. The effectiveness of this method is demonstrated by applying it to an important magnetohydrodynamics problem recently studied by Paris, Auby, and Dagazian [J. Math. Phys. 27, 2188 (1986)]. Through the very precise lower and upper bounds obtained, this approach gives full support to their analysis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528012
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Generalized Apollonian packings (1990)
    Springer
    Communications in mathematical physics 134 (1990), S. 293-319 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements ofSL(2,ℂ) as generators:R andT that are hyperbolic rotations of 2π/3 and 2π/N (N=2,3,4....), around two distinct points. The limit set of the discrete group generated byR andT provides, forN=7,8,9,.... a generalization of the Apollonian packing (which is itself recovered forN=∞). The valuesN=2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube forN=2 and 4, and the icosahedron forN=3 and 5. ForN=6 the group is no longer discrete. To further analyze this structure forN≥7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02097704
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Moment problem formulation of the Schrödinger equation (1992)
    Springer
    Numerical algorithms 3 (1992), S. 1-16 
    Details
    ISSN: 1572-9265
    Keywords: Moment problem ; Schrödinger equation ; inequalities ; convexity ; linear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We show that it is possible to project out in an exact manner the lowest eigenstate of Schrödinger equations. Taking into account the nodeless property of the lowest eigenstate one can replace the full Schrödinger equation by a moment problem whose measure is the eigenstate itself. The infinite set of positivity inequalities linked to this moment problem provides a framework which allows to compute sequences of upper and lower bounds to the unknown eigenvalue and eigenfunction. The effective computation is based on deep convexity properties embedded in the set of hierarchical inequalities associated to this moment problem. The convexity allows to get the bounds through linear programming. We illustrate the method with simple one dimensional problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02141911
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  • 4
    Electronic Resource
    Electronic Resource
    Distribution of roots of random real generalized polynomials (1997)
    Springer
    Journal of statistical physics 86 (1997), S. 675-705 
    Details
    ISSN: 1572-9613
    Keywords: Random polynomials ; density of roots ; universal asymptotic behavior
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The average density of zeros for monic generalized polynomials, $$P_n (z) = \phi (z) + \sum\nolimits_{k = 1}^n {c_k f_k } (z)$$ , with real holomorphic ϕ,f k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form $$P_n (z) = z^n + \sum\nolimits_{k = 1}^n {c_k z^{n - k} }$$ with real independent, identically distributed Gaussian coefficients having zero mean and dispersion $$\delta = 1/\sqrt {n\lambda }$$ . The average density tends to a simple,universal function of ξ=2nlog|z| and λ in the domain ξcoth(ξ/2)≪n|sin arg(z)|, where nearly all the roots are located for largen.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02199115
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  • 5
    Electronic Resource
    Electronic Resource
    Systematic construction of upper and lower bounds to the ground state energy of the Schrödinger equation (1986)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 30 (1986), S. 21-32 
    Details
    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: For the multidimensional Schrödinger equation, we develop a technique based on the moment problem formulation of this equation. The method provides systematic and rapidly converging upper and lower bounds to the ground state energy, and it enjoys the following features: 1The bounds are insensitive to the strength of the perturbation; that is, the method applies even to the most singular perturbation.2The method does not require the potential to be semibounded. It applies directly to potentials whose spectrum extends from minus to plus infinity, as is the case for the Zeeman effect in hydrogenic atoms.3The presence of continuum spectrum does not affect the method: no information coming from the continuum is needed.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560300705
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  • 6
    Electronic Resource
    Electronic Resource
    An exact projection method for the lowest eigenstate of singular multidimensional Schrödinger equations (1987)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 32 (1987), S. 755-755 
    Details
    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
    URL: http://dx.doi.org/10.1002/qua.560320777
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    Paper (German National Licenses)
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