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Semiclassical Weyl Formula for Elliptic Operators with Non-Smooth Coefficients

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Recent Advances in Operator Theory, Operator Algebras, and their Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 153))

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Abstract

We consider the Weyl formula for the asymptotic number of eigenvalues of self-adjoint elliptic differential operators with coefficients which have Hölder continuous first-order derivatives. Our aim is to prove that the Weyl formula holds with a remainder usually considered in the case of operators with smooth coefficients.

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References

  1. J. Chazarain, Spectre d’un hamiltonien quantique et mécanique classique. Comm. Partial Diff. Eq. (1980), 595–644.

    Google Scholar 

  2. M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semiclassical Limit. London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999.

    Google Scholar 

  3. B. Helfer, D. Robert, Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques. Ann. Inst. Fourier, Grenoble 31(3) (1981), 169–223.

    Google Scholar 

  4. B. Helffer, D. Robert, Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles. J. Funct. Anal. 53(3) (1983), 246–268.

    Google Scholar 

  5. L. Hörmander, The Spectral Function of an Elliptic Operator. Acta Math. 121 (1968), 173–218.

    Google Scholar 

  6. L. Hörmander, The Analysis of Linear Partial Differential Operators. Springer-Verlag, vol. 1, 2, 3, 4, Berlin-Heidelberg-New York-Tokyo, 1983, 1985.

    Google Scholar 

  7. V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics. Springer-Verlag, Berlin, 1998.

    Google Scholar 

  8. V. Ivrii, Sharp Spectral Asymptotics for Operators with Irregular Coefficients. Internat. Math. Res. Notices (2000), 1155–1166.

    Google Scholar 

  9. G. Métivier, Valeurs propres des problèmes aux limites irréguliers. Bull. Soc. Math. France Mem. 51–52 (1977), 125–219.

    Google Scholar 

  10. Y. Miyazaki, 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 

  11. Y. Miyazaki, 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 

  12. M. Reed, and B. Simon, Methods of Modern Mathematical Physics. vol. I–IV, Academic Press, New York-San Francisco-London, 1972, 1975, 1979.

    Google Scholar 

  13. D. Robert, Autour de l’approximation semi-classique. Birkhäuser, Boston 1987.

    Google Scholar 

  14. L. Zielinski, Asymptotic Behavior 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 

  15. L. Zielinski, Asymptotic Distribution of Eigenvalues for Elliptic Boundary Value Problems. Asymptot. Anal. 16 (1998), 181–201.

    Google Scholar 

  16. L. Zielinski, Asymptotic Distribution of Eigenvalues of Some Elliptic Operators with Intermediate Remainder Estimates. Asymptot. Anal. 17 (1998), 93–120.

    Google Scholar 

  17. L. Zielinski, Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-Smooth Coefficients. Math. Phys. Anal. Geom. 2 (1999), 291–321; Part 2: Colloq. Math. 92 (2002), 1–18.

    Google Scholar 

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© 2004 Birkhäuser Verlag Basel/Switzerland

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Zielinski, L. (2004). Semiclassical Weyl Formula for Elliptic Operators with Non-Smooth Coefficients. In: Gaşpar, D., Timotin, D., Zsidó, L., Gohberg, I., Vasilescu, FH. (eds) Recent Advances in Operator Theory, Operator Algebras, and their Applications. Operator Theory: Advances and Applications, vol 153. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7314-8_21

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