Adiabatic Limits and Related Lattice Point Problems

Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Let \((M,\mathcal{F})\) be a closed foliated manifold endowed with a Riemannian metric g. Then we have a direct sum decomposition \(TM\,=\,F \oplus H\) of the tangent bundle TM of M, where \(F\,=\,T\mathcal{F}\) is the tangent bundle of \(\mathcal{F}\) and \(H\,=\,{F}^{\perp }\) is the orthogonal complement of F, and the corresponding decomposition of the metric: \(g\,=\,{g}_{F} + {g}_{H}\). Consider the one-parameter family of Riemannian metrics on M,
$${g}_{\epsilon } = {g}_{F} + {\epsilon }^{-2}{g}_{ H},\quad \epsilon > 0,$$
and the corresponding Laplace-Beltrami operator \({\Delta }_{\epsilon }\). We are interested in the asymptotic behavior of the trace of the operator \(f({\Delta }_{\epsilon })\) for sufficiently nice functions f on \(\mathbb{R}\), in particular, of the eigenvalue distribution function \({N}_{\epsilon }(\lambda )\) of \({\Delta }_{\epsilon }\), as \(\epsilon \rightarrow 0\) (in the adiabatic limit).

References

  1. 1.
    J. Álvarez López, Yu. A. Kordyukov, Adiabatic limits and spectral sequences for Riemannian foliations, Geom. and Funct. Anal. 10 (2000), 977–1027.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Yu. A. Kordyukov, Quasiclassical asymptotics of the spectrum of elliptic operators in a foliated manifold. Math. Notes 53 (1993), no. 1–2, 104–105.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Yu. A. Kordyukov, On the quasiclassical asymptotics of the spectrum of hypoelliptic operators on a manifold with foliation. Funct. Anal. Appl. 29 (1995), no. 3, 211–213.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Yu. A. Kordyukov, Adiabatic limits and spectral geometry of foliations, Math. Ann. 313 (1999), 763–783.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Yu. A. Kordyukov, Semiclassical spectral asymptotics on foliated manifolds, Math. Nachr. 245 (2002), 104–128.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Yu. A. Kordyukov, A. A. Yakovlev, Adiabatic limits and the spectrum of the Laplacian on foliated manifolds, C  ∗ -algebras and elliptic theory. II, Trends in Mathematics, Birkhäuser, Basel, 2008, 123–144.Google Scholar
  7. 7.
    Yu. A. Kordyukov, A. A. Yakovlev, Integer points in domains and adiabatic limits, Algebra i Analiz, 23 (2011), no. 6, 80–95; preprint arXiv:1006.4977.Google Scholar
  8. 8.
    Yu. A. Kordyukov, A. A. Yakovlev, The number of integer points in a family of anisotropically expanding domains, in preparation.Google Scholar
  9. 9.
    Nikichine, N. A.; Skriganov, M. M., Nombre de points d’un réseau dans un produit cartésien de domaines convexes, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 671–675.MathSciNetMATHGoogle Scholar
  10. 10.
    Nikishin, N. A.; Skriganov, M. M.. On the distribution of algebraic numbers in parallelotopes, St. Petersburg Math. J. 10 (1999), no. 1, 53–68MathSciNetGoogle Scholar
  11. 11.
    A. A. Yakovlev, Adiabatic limits on Riemannian Heisenberg manifolds, Sb. Math. 199 (2008), no. 1–2, 307–318.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    A. A. Yakovlev, Asymptotics of the spectrum of the Laplace operator on Riemannian Sol-manifolds in the adiabatic limit, Sib. Math. J. 51 (2010), no. 2, 370–382.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Institute of MathematicsRussian Academy of SciencesUfaRussia

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