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Spectral asymptotics of differential and pseudodifferential operators. II

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Abstract

In the article we obtain asymptotic formulae with remainder estimates for the distribution function of the eigenvalues of degenerate elliptic differential operators and Schrödinger operators with singular potential.

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Literature cited

  1. K. Kh. Boimatov, “Spectral asymptotic of differential and pseudodifferential operators. I,” in: Tr. Sem. Petrovsk.,7, 1981.

  2. A. G. Kostyuchenko, “Asymptotic behavior of the spectral function of self-adjoint elliptic operators,” in: Chetvertaya Mat. Shkola, Naukova Dumka, Kiev (1968).

    Google Scholar 

  3. M. G. Gasymov, “On the distribution of the eigenvalues of a self-adjoint ordinary differential operator,” Dokl. Akad. Nauk SSSR,186, No. 4, 753–756 (1969).

    Google Scholar 

  4. K. Kh. Boimatov, “Pseudodifferential operators with an operator symbol,” Dokl. Akad. Nauk SSSR,244, No. 1, 20–23 (1979).

    Google Scholar 

  5. K. Kh. Boimatov, “Spectral asymptotics of differential equations with operator coefficients in a nonsmooth domain,” Dokl. Akad. Nauk SSSR,242, No. 4, 749–752 (1978).

    Google Scholar 

  6. K. Kh. Boimatov, “The asymptotic behavior of the spectrum of the Schrödinger operator with singular potential,” Usp. Mat. Nauk,31, No. 1, 241–242 (1976).

    Google Scholar 

  7. K. Kh. Boimatov, “The asymptotic behavior of the spectrum of an elliptic differential operator in Rn in the degenerate case,” Dokl. Akad. Nauk SSSR,243, No. 6, 1369–1372 (1978).

    Google Scholar 

  8. K. Kh. Boimatov, “The distribution of the eigenvalues of degenerate operators,” Dokl. Akad. Nauk SSSR,248, No. 3, 521–524 (1979).

    Google Scholar 

  9. M. Sh. Birman and M. Z. Solomyak, “The asymptotic behavior of the spectra of differential equations,” J. Sov. Math.,12, No. 3 (1979).

  10. V. I. Bezyaev, “The asymptotics of the eigenvalues of hypoelliptic operators on a closed manifold,” Dokl. Akad. Nauk SSSR,244, No. 5, 1054–1057 (1979).

    Google Scholar 

  11. V. B. Moscatelli and M. Thompson, “Distribution asymptotique des valeurs propres d'operateurs pseudo-differentiels sur des varietes compactes,” C. R. Acad. Sci. Paris,244, No. 6, A373-A375 (1975).

    Google Scholar 

  12. A. Manikof and J. Sjostrand, “On the eigenvalues of a class of hypoelliptic operators,” Math. Ann.235, No. 1, 55–85 (1978).

    Google Scholar 

  13. L. A. Pastur, “Spectra of random self-adjoint operators,” Usp. Mat. Nauk,28, No. 1, 3–64 (1973).

    Google Scholar 

  14. M. A. Shubin, “Elliptic almost-periodic operators and von Neumann algebras,” Funkts. Anal. Prilozhen.,9, No. 1, 89–90 (1975).

    Google Scholar 

  15. M. A. Shubin, “On the asymptotics of the spectrum of elliptic operators with almost-periodic coefficients,” Usp. Mat. Nauk,30, No. 2, 268–269 (1975).

    Google Scholar 

  16. M. A. Shubin, “Differential and pseudodifferential operators in spaces of almost-periodic functions,” Mat. Sb.,95, No. 4, 560–587 (1974).

    Google Scholar 

  17. M. A. Shubin, “Pseudodifferential almost-periodic operators and von Neumann algebras,” Tr. Mosk. Mat. Obshch.,35, 103–163 (1976).

    Google Scholar 

  18. M. A. Shubin, “The density of states of self-adjoint elliptic operators with almost-periodic coefficients,” Tr. Sem. Petrovsk.,3, 243–275 (1978).

    Google Scholar 

  19. V. Yu. Kiselev, “Almost-periodic Fourier integral operators and some of their applications,” Tr. Sem. Petrovsk.,3, 81–97 (1977).

    Google Scholar 

  20. V. I. Bezyaev, “The asymptotic behavior of the density of states of hypoelliptic almost-periodic operators,” Mat. Sb.,105, No. 4, 485–511 (1978).

    Google Scholar 

  21. M. A. Shubin, “Weyl's theorem for the Schrödinger operator with an almost-periodic potential,” Vestn. Mosk. Univ. Ser. Mat. Mekh.,1976, No. 2, 84–88.

    Google Scholar 

  22. M. A. Shubin, “Almost-periodic functions and partial differential operators,” Usp. Mat. Nauk,33, No. 2, 3–47 (1978).

    Google Scholar 

  23. M. A. Shubin, “The spectral theory and the index of elliptic operators with almost-periodic coefficients,” Usp. Mat. Nauk,34, No. 2, 95–135 (1979).

    Google Scholar 

  24. B. Ya. Skachek, “The distribution of the eigenvalues of multidimensional differential operators,” Funkts. Anal. Prilozhen.,9, No. 1, 83–84 (1975).

    Google Scholar 

  25. G. Metivier, “Comportement asymptotique des propres d'operateurs elliptiques degeneres,” Asterisque,34–35, 215–216 (1976).

    Google Scholar 

  26. M. Sh. Birman and M. Z. Solomyak, “Spectral asymptotics of nonsmooth elliptic operators. I,” Tr. Mosk. Mat. Obshch.,27, 3–52 (1972).

    Google Scholar 

  27. M. Sh. Birman and M. Z. Solomyak, “Spectral asymptotics of nonsmooth elliptic operators. II,” Tr. Mosk. Mat. Obshch.,28, 3–34 (1973).

    Google Scholar 

  28. H. Triebel, “Lp-theory for a class of singular elliptic differential operators,” Czechoslovak Math. J.,23, No. 4, 525–541 (1973).

    Google Scholar 

  29. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1970).

  30. H. Triebel, “Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam (1978).

    Google Scholar 

  31. L. Hormander, Linear Partial Differential Operators, Springer, Berlin-Heidelberg-New York-Tokyo (1963).

    Google Scholar 

  32. N. Nilsson, “Some estimates for spectral functions connected with formally hypoelliptic differential operators,” Ark. Mat.,10, No. 2, 251–275 (1972).

    Google Scholar 

  33. M. Otelbaev and Ya. T. Sultanaev, “On the formulas for the distribution of eigenvalues of singular differential operators,” Mat. Zametki,14, No. 3, 361–368 (1974).

    Google Scholar 

  34. G. V. Rozenblyum, “The asymptotic behavior of the eigenvalues of the Schrödinger operator,” Mat. Sb.,93, No. 3, 346–367 (1974).

    Google Scholar 

  35. I. A. Kipriyanov, “The asymptotic distribution of eigenvalues and eigenfunctions of a certain class of singular elliptic operators,” Tr. Mat. Inst. Akad. Mauk SSSR,117, 159–179 (1972).

    Google Scholar 

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Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 78–106, 1984.

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Boimatov, K.K. Spectral asymptotics of differential and pseudodifferential operators. II. J Math Sci 35, 2744–2769 (1986). https://doi.org/10.1007/BF01119189

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  • DOI: https://doi.org/10.1007/BF01119189

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