Journal of Mathematical Sciences

, Volume 128, Issue 2, pp 2822–2824 | Cite as

Integral Equations of Fredholm Type with Rapidly Varying Kernels and Their Relationship to Dynamic Systems

  • S. Yu. Slavyanov


The relationship between the eigenfunctions of a Fredholm-type integral equation with rapidly oscillating kernel and the dynamic mapping is analyzed. Differential operators commuting with the Fourier operator are constructed. These operators are closely related to nontrivial solutions of the unperturbed nonlinear functional equation related to the dynamic mapping. Bibliography: 6 titles.


Fourier Dynamic System Integral Equation Functional Equation Differential Operator 
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  1. 1.
    V. F. Lazutkin, “Spectral degeneracy and “small denominators” in the asymptotics of eigenfunctions of “bouncing ball” type,” Vestnik Leningrad. Univ., 24, No.7, 23–34 (1969).Google Scholar
  2. 2.
    V. F. Lazutkin, “Eigenfunctions with given caustic,” Zh. Vychisl. Mat. Mat. Fiz., 10, No.2, 362–373 (1970).Google Scholar
  3. 3.
    S. Yu. Slavyanov, “The construction of the asymptotics of the eigenfunctions of Fredholm type integral equations with symmetric rapidly oscillating kernels,” in: Problems of Mathematical Physics [in Russian], No. 6, Izdat. Leningrad. Univ., Leningrad (1973), pp. 134–141.Google Scholar
  4. 4.
    S. Yu. Slavyanov, “On the theory of open resonators,” Zh. Exper. Teor. Fiz., 64, 785–795 (1973).Google Scholar
  5. 5.
    S. Yu. Slavyanov and V. G. Farafonov, “Aberrations in open resonators with axial symmetry,” Opt. & Spectr., 76, 182–184 (1979).Google Scholar
  6. 6.
    L. I. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions [in Russian], Nauka, Moscow (1976).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. Yu. Slavyanov
    • 1
  1. 1.Department of Computational PhysicsSt.Petersburg State UniversitySt. PetersburgRussia

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