Skip to main content
SpringerLink
Log in

Two-term asymptotics of the spectrum of a boundary-value problem under an interior reflection of general form

  • Published:
Functional Analysis and Its Applications Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature Cited

  1. J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vols. I–III, Springer-Verlag, New York (1972–1973).

    Google Scholar 

  2. J. J. Duistermaat and V. W. Guillemin, "The spectrum of positive elliptic operators and periodic bicharacteristics," Invent. Math.,29, No. 1, 39–79 (1975).

    Google Scholar 

  3. D. G. Vasil'ev, "Two-term asymptotics of the spectrum of a boundary-value problem," Funkts. Anal. Prilozhen.,17, No. 4, 79–81 (1983).

    Google Scholar 

  4. I. P. Kornfel'd, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  5. V. S. Buslaev, "Scattered plane waves, spectral asymptotics, and trace formulas in exterior problems," Dokl. Akad. Nauk SSSR,197, No. 5, 999–1002 (1971).

    Google Scholar 

  6. B. M. Levitan, "On the expansion in eigenfunctions of the Laplace operator," Mat. Sb.,35, No. 2, 267–316 (1954).

    Google Scholar 

  7. S. Agmon, "Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators," Arch. Rational Mech. Anal.,28, No. 3, 165–183 (1968).

    Google Scholar 

  8. L. Hörmander, "The spectral function of an elliptic operator," Acta Math.,121, Nos. 1–2, 193–218 (1968).

    Google Scholar 

  9. J. Brüning, "Zur Abschätzung der Spektralfunktion elliptischer Operatoren," Math. Z.,137, No. 1, 75–85 (1974).

    Google Scholar 

  10. D. Robert, "Developpement asymptotique du noyau resolvant d'operateurs elliptiques," Osaka J. Math.,15, No. 2, 233–243 (1978).

    Google Scholar 

  11. R. Seeley, "A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of R3," Adv. Math.,29, No. 2, 244–269 (1978).

    Google Scholar 

  12. R. Seeley, "An estimate near the boundary for the spectral function of the Laplace operator," Am. J. Math.,102, No. 5, 869–902 (1980).

    Google Scholar 

  13. V. Ya. Ivrii, "Second term of the spectral asymptotics for the Laplace—Beltrami operator on manifolds with boundary," Funkts. Anal. Prilozhen.,14, No. 2, 25–34 (1980).

    Google Scholar 

  14. V. Ya. Ivrii, "On the exact spectral asymptotics for the Laplace—Beltrami operator under general elliptic boundary conditions," Funkts. Anal. Prilozhen.,15, No. 1, 74–75 (1981).

    Google Scholar 

  15. V. Ya. Ivrii, "The asymptotic behavior of the eigenvalues for certain elliptic operators acting in vector bundles over a manifold with boundary," Dokl. Akad. Nauk SSSR,258, No. 5, 1045–1046 (1981).

    Google Scholar 

  16. V. Ya. Ivrii, "The asymptotics of a certain spectral problem associated with the Laplace—Beltrami operatoron a manifold with boundary," Funkts. Anal. Prilozhen.,17, No. 1, 71–72 (1983).

    Google Scholar 

  17. M. A. Shubin, Pseudodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  18. L. Hörmander, "Fourier integral operators. I," Acta Math.,127, Nos. 1–2, 79–183 (1971).

    Google Scholar 

  19. F. Treves, Introduction to Pseudodifferential Operators and Fourier Integral Operators, Vols. 1 and 2, Plenum Press, New York (1980).

    Google Scholar 

  20. M. V. Fedoryuk, "Singularities of the kernels of Fourier integral operators, and the asymptotic behavior of the solution of a mixed problem," Usp. Mat. Nauk,32, No. 6, 67–115 (1977).

    Google Scholar 

  21. M. S. Agranovich and M. I. Vishik, "Elliptic problems with a parameter, and parabolic problems of general form," Usp. Mat. Nauk,19, No. 3, 53–161 (1964).

    Google Scholar 

  22. B. M. Levitan, "The asymptotic behavior of the spectral function of an elliptic equation," Usp. Mat. Nauk,26, No. 6, 151–212 (1971).

    Google Scholar 

  23. L. D. Landau and E. M. Lifshits (Lifshitz), Theory of Elasticity, Pergamon Press, London (1959).

    Google Scholar 

  24. P. Debye, "Zur Theorie der spezifischen Wärmen," Ann. Phys.,39, 789–839 (1912).

    Google Scholar 

  25. M. Dupuis, R. Mazo, and L. Onsager, "Surface specific heat of an isotropic solid at low temperatures," J. Chem. Phys.,33, No. 5, 1452–1461 (1960).

    Google Scholar 

  26. G. Metivier, "Estimation du reste en theorie spectrale," in: Seminar Analysis 1982/83, Inst. Math., Berlin (1983), pp. 70–96.

    Google Scholar 

Download references

Authors
  1. D. G. Vasil'ev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Institute for Problems of Mechanics, Academy of Sciences of the USSR. Translatéd from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 18, No. 4, pp. 1–13, October–December. 1984.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vasil'ev, D.G. Two-term asymptotics of the spectrum of a boundary-value problem under an interior reflection of general form. Funct Anal Its Appl 18, 267–277 (1984). https://doi.org/10.1007/BF01083689

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01083689

Keywords

Search

Navigation