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  • 34B25  (2)
  • 43A90  (1)
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  • 1
    Electronic Resource
    Electronic Resource
    On the ubiquitous Gelfand-Levitan-Marčenko (GLM) equation (1990)
    Springer
    Acta applicandae mathematicae 18 (1990), S. 99-144 
    Details
    ISSN: 1572-9036
    Keywords: 22E70 ; 30E25 ; 33A75 ; 34A55 ; 34B25 ; 35G20 ; 35P10 ; 35P25 ; 47G05 ; 49H05 ; 53B99 ; 58F05 ; 70H05 ; 76Z99 ; 81F99 ; Gelfand-Levitan-Marčenko equations ; Riemann-Hilbert problem ; coadjoint orbits ; tau functions ; Kadomtsev-Petviashvili equations ; hlerarchies ; direct linearization ; inverse scattering ; generalized eigenfunctions ; wave functions ; Korteweg-deVries equation ; vertex operators
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We collect a number of derivations and interpretations of Gelfand-Levitan-Marčenko (GLM) equations in various contexts. Relations to Riemann-Hilbert (RH) problems, tau functions and determinants, direct linearization techniques, etc. are discussed and some new results are included. We emphasize the role of spectral pairings of generalized eigenfunctions and exhibit various ways in which inverse scattering techniques arise. The natural extension and generalization of GLM techniques in the format of RH methods and coadjoint orbits in a Lie theoretic context is developed and the hierarchy point of view is explored in terms of spectral ideas. Thus, the roles and some limitations of the GLM and RH methods are exposed to some extent and some bridges are sketched to modern techniques in soliton mathematics involving hierarchies, tau functions, Lie theory, etc.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00046563
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  • 2
    Electronic Resource
    Electronic Resource
    Patterns and structure in systems governed by linear second-order differential equations (1986)
    Springer
    Acta applicandae mathematicae 6 (1986), S. 109-184 
    Details
    ISSN: 1572-9036
    Keywords: 33A45 ; 34A55 ; 34B25 ; 35A22 ; 35P10 ; 35P25 ; 35R30 ; 42A38 ; 42A85 ; 43A90 ; 44A05 ; 45B05 ; 45D05 ; 45E10 ; 47A68 ; 47B35 ; 60G35 ; 60G60 ; 62M15 ; 62M20 ; 73D30 ; 81F15 ; 86A15 ; 93E10 ; 93E11 ; Transmutation ; transformation operators ; Parseval formulas ; eigenfunction expansions ; Paley-Wiener theory ; Riemann-Liouville and Weyl integrals ; special functions ; Gelfand-Levitan and Marčenko equations ; generalized translation ; generalized convolution ; Volterra operators ; Toeplitz operators ; Wiener-Hopf equations ; Bergman-Gilbert operators ; Kontorovič-Lebedev inversion ; generalized axially symmetric potential theory ; canonical equations ; deBranges spaces ; random evolutions ; scattering theory ; inverse problems ; filtering ; smoothing ; innovations ; Darboux-Christoffel formulas ; generating functions ; reproducing kernels ; random fields ; linear stochastic estimation ; spectral measures ; orthogonal polynomials ; least squares ; etc
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This article represents a survey of transmutation ideas and their interaction with typical physical problems. For linear second-order differential operatorsP andQ one deals with canonical connectionsB:P→Q (transmutations) satisfyingQB=BP and the related transport of ‘structure’ between the theories ofP andQ. One can study in an intrinsic manner, e.g., Parseval formulas, eigenfunction expansions, integral transform, special functions, inverse problems, integral equations, and related stochastic filtering and estimation problems, etc. There are applications in virtually any area where such operators arise.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00046724
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    Paper (German National Licenses)
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