Asymptotic Distribution of Eigenvalues for a Class of Second-Order Elliptic Operators with Irregular Coefficients in Rd

  • Lech Zielinski
Article

Abstract

Let A=A0+v(x) where A0 is a second-order uniformly elliptic self-adjoint operator in Rd and v is a real valued polynomially growing potential. Assuming that v and the coefficients of A0 are Hölder continuous, we study the asymptotic behaviour of the counting function N(A,λ) (λ→∞) with the remainder estimates depending on the regularity hypotheses. Our strongest regularity hypotheses involve Lipschitz continuity and give the remainder estimate N(A,λ)O({λ}−μ), where μ may take an arbitrary value strictly smaller than the best possible value known in the smooth case. In particular, our results are obtained without any hypothesis on critical points of the potential.

spectral asymptotics Weyl formula Schrödinger operator elliptic operator pseudodifferential operator 

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References

  1. 1.
    Birman, M. and Solomyak, M.: Asymptotic behaviour of spectrum of differential equations, J. Soviet. Math. 12 (1979), 247–282.Google Scholar
  2. 2.
    Boimatov, K.: Spectral asymptotics of pseudodifferential operators, Soviet Math. Dokl. 42(2) (1990), 196–200.Google Scholar
  3. 3.
    Buzano, E.: Some remarks on the Weyl asymptotics by the approximate spectral projection method, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. 8 (2000), 775–792.Google Scholar
  4. 4.
    Dencker, N.: The Weyl calculus with locally temperate metrics and weights, Ark. Mat. 24 (1986), 59–79.Google Scholar
  5. 5.
    Edmunds, D. E. and Evans, W. D.: On the distribution of eigenvalues of Schrödinger operators, Arch. Rational Mech. Anal. 89 (1985), 135–167.Google Scholar
  6. 6.
    Edmunds, D. E. and Evans, W. D.: Spectral Theory and Differential Operators, Oxford Math. Monogr., Oxford, 1989.Google Scholar
  7. 7.
    Faris, W. G.: Self-adjoint Operators, Lecture Notes in Math. 433, Springer-Verlag, New York, 1975.Google Scholar
  8. 8.
    Feigin, V. I.: The asymptotic distribution of the eigenvalues of pseudodifferential operators in R n, Math. USSR-Sb. 28 (1976), 533–552.Google Scholar
  9. 9.
    Feigin, V. I.: Sharp estimates of the remainder in the spectral asymptotics for pseudodifferential operators in R n, Functional Anal. Appl. 16 (1982), 88–89.Google Scholar
  10. 10.
    Fleckinger, J. and Lapidus, M.: Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights, Arch. Rational Mech. Anal. 98 (1987), 329–356.Google Scholar
  11. 11.
    Fleckinger, J. and Lapidus, M.: Schrödinger operators with indefinite weight functions: asymptotics of eigenvalues with remainder estimates, Differential Integral Equations 7 (1994), 1389–1418.Google Scholar
  12. 12.
    Guillemin, V. and Sternberg, S.: Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), 915–955.Google Scholar
  13. 13.
    Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque 112 (1984).Google Scholar
  14. 14.
    Helffer, B. and Robert, D.: Propriétés asymptotiques du spectre d'opérateurs pseudodifférentiels sur R n, Comm. Partial Differential Equations 7 (1982), 795–881.Google Scholar
  15. 15.
    Hörmander, L.: On the asymptotic distribution of the pseudodifferential operators in R n, Ark. Mat. 17 (1979), 297–313.Google Scholar
  16. 16.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols 1–4, Springer-Verlag, New York, 1983, 1985.Google Scholar
  17. 17.
    Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, Berlin, 1998.Google Scholar
  18. 18.
    Ivrii, V.: Sharp spectral asymptotics for operators with irregular coefficients, Internat. Math. Res. Notices (2000), 1155–1166.Google Scholar
  19. 19.
    Kumano-Go, H.: Pseudodifferential Operators, MIT Press, Cambridge, MA, 1981.Google Scholar
  20. 20.
    Kumano-Go, H. and Nagase, M.: Pseudodifferential operators with non-regular symbols and applications, Funkcial. Ekvac. 21 (1978), 151–192.Google Scholar
  21. 21.
    Levendorskii, S. Z.: Asymptotic Distribution of Eigenvalues of Differential Operators, Math. Appl., Kluwer Acad. Publ., Dordrecht, 1990.Google Scholar
  22. 22.
    Métivier, G.: Valeurs propres des problè mes aux limites irréguliers, Bull. Soc. Math. France Mem. 51–52 (1977), 125–219.Google Scholar
  23. 23.
    Miyazaki, Y.: A sharp asymptotic remainder estimate for the eigenvalues of operators associated with strongly elliptic sesquilinear forms, Japan. J. Math. 15 (1989), 65–97.Google Scholar
  24. 24.
    Miyazaki, Y.: The eigenvalue distribution of elliptic operators with Hölder continuous coefficients, Osaka J. Math. 28 (1991), 935–973; Part 2, Osaka J. Math. 30 (1993), 267–302.Google Scholar
  25. 25.
    Mohamed, A.: Comportement asymptotique avec estimation du reste, des valeurs propres d'une classe d'opérateurs pseudo-différentiels sur R n, Math. Nachr. 140 (1989), 127–186.Google Scholar
  26. 26.
    Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, vols I-IV, Academic Press, New York, 1972, 1975, 1979.Google Scholar
  27. 27.
    Robert, D.: Propriétés spectrales d'opérateurs pseudo-différentiels, Comm. Partial Differential Equations 3 (1978), 755–826.Google Scholar
  28. 28.
    Rozenblyum, G. V.: Asymptotics of the eigenvalues of the Schrödinger operator, Math. USSRSb. 22 (1974), 349–371.Google Scholar
  29. 29.
    Shubin, M. A. and Tulovskii, V. A.: On the asymptotic distribution of eigenvalues of pseudodifferential operators in R n, Math. USSR-Sb. 21 (1973), 565–573.Google Scholar
  30. 30.
    Tamura, H.: Asymptotic formula with remainder estimates for eigenvalues of Schrödinger operators, Comm. Partial Differential Equations 7 (1982), 1–54.Google Scholar
  31. 31.
    Tamura, H.: Asymptotic formula with sharp remainder estimates for eigenvalues of elliptic operators of second order, Duke Math. J. 49 (1982), 87–119.Google Scholar
  32. 32.
    Zielinski, L.: Asymptotic behaviour of eigenvalues of differential operators with irregular coefficients on a compact manifold, C.R. Acad. Sci. Paris Sér. I Math. 310 (1990), 563–568.Google Scholar
  33. 33.
    Zielinski, L.: Asymptotic distribution of eigenvalues for elliptic boundary value problems, Asymptotic Anal. 16 (1998), 181–201.Google Scholar
  34. 34.
    Zielinski, L.: Asymptotic distribution of eigenvalues of some elliptic operators with intermediate remainder estimates, Asymptotic Anal. 17 (1998), 93–120.Google Scholar
  35. 35.
    Zielinski, L.: Sharp spectral asymptotics and Weyl formula for elliptic operators with nonsmooth coefficients, Math. Phys. Anal. Geom. 2 (1999), 291–321; Part 2, Colloq. Math. 92 (2002), 1–18.Google Scholar

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© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Lech Zielinski
    • 1
    • 2
  1. 1.LMPAUniversité du LittoralCalais CedexFrance
  2. 2.IMJ, Mathématiques, case 7012Université Paris 7Paris Cedex 05France

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