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.
Similar content being viewed by others
Literature cited
K. Kh. Boimatov, “Spectral asymptotic of differential and pseudodifferential operators. I,” in: Tr. Sem. Petrovsk.,7, 1981.
A. G. Kostyuchenko, “Asymptotic behavior of the spectral function of self-adjoint elliptic operators,” in: Chetvertaya Mat. Shkola, Naukova Dumka, Kiev (1968).
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).
K. Kh. Boimatov, “Pseudodifferential operators with an operator symbol,” Dokl. Akad. Nauk SSSR,244, No. 1, 20–23 (1979).
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).
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).
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).
K. Kh. Boimatov, “The distribution of the eigenvalues of degenerate operators,” Dokl. Akad. Nauk SSSR,248, No. 3, 521–524 (1979).
M. Sh. Birman and M. Z. Solomyak, “The asymptotic behavior of the spectra of differential equations,” J. Sov. Math.,12, No. 3 (1979).
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).
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).
A. Manikof and J. Sjostrand, “On the eigenvalues of a class of hypoelliptic operators,” Math. Ann.235, No. 1, 55–85 (1978).
L. A. Pastur, “Spectra of random self-adjoint operators,” Usp. Mat. Nauk,28, No. 1, 3–64 (1973).
M. A. Shubin, “Elliptic almost-periodic operators and von Neumann algebras,” Funkts. Anal. Prilozhen.,9, No. 1, 89–90 (1975).
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).
M. A. Shubin, “Differential and pseudodifferential operators in spaces of almost-periodic functions,” Mat. Sb.,95, No. 4, 560–587 (1974).
M. A. Shubin, “Pseudodifferential almost-periodic operators and von Neumann algebras,” Tr. Mosk. Mat. Obshch.,35, 103–163 (1976).
M. A. Shubin, “The density of states of self-adjoint elliptic operators with almost-periodic coefficients,” Tr. Sem. Petrovsk.,3, 243–275 (1978).
V. Yu. Kiselev, “Almost-periodic Fourier integral operators and some of their applications,” Tr. Sem. Petrovsk.,3, 81–97 (1977).
V. I. Bezyaev, “The asymptotic behavior of the density of states of hypoelliptic almost-periodic operators,” Mat. Sb.,105, No. 4, 485–511 (1978).
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.
M. A. Shubin, “Almost-periodic functions and partial differential operators,” Usp. Mat. Nauk,33, No. 2, 3–47 (1978).
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).
B. Ya. Skachek, “The distribution of the eigenvalues of multidimensional differential operators,” Funkts. Anal. Prilozhen.,9, No. 1, 83–84 (1975).
G. Metivier, “Comportement asymptotique des propres d'operateurs elliptiques degeneres,” Asterisque,34–35, 215–216 (1976).
M. Sh. Birman and M. Z. Solomyak, “Spectral asymptotics of nonsmooth elliptic operators. I,” Tr. Mosk. Mat. Obshch.,27, 3–52 (1972).
M. Sh. Birman and M. Z. Solomyak, “Spectral asymptotics of nonsmooth elliptic operators. II,” Tr. Mosk. Mat. Obshch.,28, 3–34 (1973).
H. Triebel, “Lp-theory for a class of singular elliptic differential operators,” Czechoslovak Math. J.,23, No. 4, 525–541 (1973).
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1970).
H. Triebel, “Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam (1978).
L. Hormander, Linear Partial Differential Operators, Springer, Berlin-Heidelberg-New York-Tokyo (1963).
N. Nilsson, “Some estimates for spectral functions connected with formally hypoelliptic differential operators,” Ark. Mat.,10, No. 2, 251–275 (1972).
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).
G. V. Rozenblyum, “The asymptotic behavior of the eigenvalues of the Schrödinger operator,” Mat. Sb.,93, No. 3, 346–367 (1974).
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).
Additional information
Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 78–106, 1984.
Rights and permissions
About this article
Cite this article
Boimatov, K.K. Spectral asymptotics of differential and pseudodifferential operators. II. J Math Sci 35, 2744–2769 (1986). https://doi.org/10.1007/BF01119189
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01119189