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Functional Analysis and Its Applications

, Volume 17, Issue 3, pp 193–200 | Cite as

Asymptotic expansion of the spectral function for second-order elliptic operators in Rn

  • G. S. Popov
  • M. A. Shubin
Article

Keywords

Functional Analysis Asymptotic Expansion Spectral Function Elliptic Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature Cited

  1. 1.
    V. S. Buslaev, "On the asymptotic behavior of spectral characteristics of exterior problems for the Schrödinger operator," Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 1, 148–235 (1975).Google Scholar
  2. 2.
    A. A. Arsen'ev, "Asymptotic behavior of the spectral function of the Schrödinger equation," Zh. Vychisl. Mat. Mat. Fiz.,7, No. 6, 507–518 (1967).Google Scholar
  3. 3.
    B. R. Vainberg, "On the short-wave asymptotic of solutions to stationary problems and the asymptotic for t → ∞ of solutions to nonstationary problems," Usp. Mat. Nauk,30, No. 2, 1–55 (1975).Google Scholar
  4. 4.
    V. S. Buslaev, "Scattered plane waves, spectral asymptotics, and trace formulas for exterior problems," Dokl. Akad. Nauk SSSR,197, No. 5, 999–1002 (1971).Google Scholar
  5. 5.
    A. Majda and J. Ralston, "An analogue of Weyl's formula for unbounded domains. I, II, III," Duke Math. J.,45, No. 1, 183–196 (1978);45, No. 3, 513–536;46, No. 4, 725–731 (1979).Google Scholar
  6. 6.
    V. Petkov and G. Popov, "Asymptotic behavıor of the scattering phase for nontrapping obstacles," Ann. Inst. Fourier,32, No. 3, 111–150 (1982).Google Scholar
  7. 7.
    V. Ya. Ivrii and M. A. Shubin, "On the asymptotic behavior spectral shift function," Dokl. Akad. Nauk SSSR,263, No. 2, 283–284 (1982).Google Scholar
  8. 8.
    L. Hörmander, "The spectral function of an elliptic operator," Acta Math.,121, Nos. 3–4, 193–218 (1968).Google Scholar
  9. 9.
    J. Rauch, "Asymptotic behaviour of solutions to hyperbolic differential equations with zero speeds," Commun. Pure Appl. Math.,31, No. 4, 431–480 (1978).Google Scholar
  10. 10.
    J. Duistermaat and L. Hörmander, "Fourier integral operators. II," Acta Math.,128, Nos. 3–4, 183–269 (1972).Google Scholar
  11. 11.
    S. Kuroda, "Scattering theory for differential operators. I. Operator theory. II. Self-adjoint elliptic operators," J. Math. Soc. Jpn.,25, No. 1, 75–104; No. 4, 222–234 (1973).Google Scholar
  12. 12.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Academic Press, New York (1972).Google Scholar
  13. 13.
    H. McKean and I. Singer, "Curvature and the eigenvalues of the Laplacian," J. Diff. Geom.,1, No. 4, 43–69 (1967).Google Scholar
  14. 14.
    J. Duistermaat and V. Guillemin, "The spectrum of positive elliptic operators and periodic bicharacteristics," Invent. Math.,29, No. 1, 39–79 (1975).Google Scholar
  15. 15.
    V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys, No. 14, Amer. Math. Soc., Providence, Rhode Island (1977).Google Scholar
  16. 16.
    M. A. Shubin, Pseudodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1978).Google Scholar
  17. 17.
    V. M. Babich, "On the short-wave asymptotic of the solution to the point-source problem in a nonhomogeneous medium," Zh. Vychisl. Mat. Mat. Fiz.,5, No. 5, 949–951 (1965).Google Scholar
  18. 18.
    V. V. Kucherenko. "Quasiclassical asymptotics of the point-source function for the stationary Schrödinger equation," Theor. Mat. Fiz.,1, No. 3, 384–406 (1969).Google Scholar
  19. 19.
    V. V. Kucherenko, "Some properties of the short-wave asymptotic of the fundamental solution of the equation [Δ + k2n(x)]u = 0," in: Trudy MIÉM, Asymptotic Methods and Difference Schemes, Vol. 25, Moscow (1972).Google Scholar
  20. 20.
    V. M. Babich and Yu. O. Rapoport, "Short-time asymptotic of the fundamental solution to the Cauchy problem for a second-order parabolic equation," in: Problems of Mathematical Physics, No. 7, Leningrad State Univ. (1974), pp. 21–38.Google Scholar
  21. 21.
    V. M. Babich, "Hadamard's method and the asymptotic of the spectral function of a secondorder differential operator," Mat. Zametki,28, No. 5, 689–694 (1980).Google Scholar
  22. 22.
    G. S. Popov and M. A. Shubin, "Complete asymptotic expansion of the spectral function for second-order elliptic operators in Rn," Usp. Mat. Nauk,38, No. 1, 187–188 (1983).Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • G. S. Popov
  • M. A. Shubin

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