Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators

  • Victor IvriiEmail author


Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function \(e_{h,\varepsilon }(x, x,\lambda )\) for a scalar operator
$$\begin{aligned} A_\varepsilon (x, hD)= A^0(hD) + \varepsilon B(x, hD), \end{aligned}$$
where \(A^0\) is an elliptic operator and B(xhD) is a periodic or almost periodic perturbation.

In particular, a complete semiclassical asymptotics of the integrated density of states also holds. Further, we consider generalizations.

Key words and phrases

Microlocal Analysis sharp spectral asymptotics integrated density of states periodic and almost periodic operators Diophantine conditions 

2010 Mathematics Subject Classification:



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  1. [BH]
    A. Barnett; A. Hassell. Fast computation of high-frequency Dirichlet eigenmodes via spectral flow of the interior Neumann-to-Dirichlet map. Comm. Pure Appl. Math. 67 (2014), no. 3, 351–407.MathSciNetCrossRefGoogle Scholar
  2. [BFK]
    D. Burghelea; L. Friedlander; T. Kappeler. Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107 (1992), no. 1, 34–65.MathSciNetCrossRefGoogle Scholar
  3. [Cal]
    A. P. Calderón. On an inverse boundary value problem Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp. 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980.Google Scholar
  4. [DGHH]
    K. Datchev; J. Gell-Redman; A. Hassell; P. Humphries. Approximation and equidistribution of phase shifts: spherical symmetry. Comm. Math. Phys. 326 (2014), no. 1, 209–236.MathSciNetCrossRefGoogle Scholar
  5. [DuGu]
    J. J. Duistermaat; V. W. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1975), no. 1, 39–79.MathSciNetCrossRefGoogle Scholar
  6. [FG]
    R. L. Frank; L. Geisinger. Semi-classical analysis of the Laplace operator with Robin boundary conditions. Bull. Math. Sci. 2 (2012), no. 2, 281–319.MathSciNetCrossRefGoogle Scholar
  7. [Fried]
    L. Friedlander. Some inequalities between Dirichlet and Neumann eigenvalues. Arch. Rational Mech. Anal. 116 (1991), no. 2, 153–160.MathSciNetCrossRefGoogle Scholar
  8. [GHZ]
    J. Gell-Redman; A. Hassell; S. Zelditch. Equidistribution of phase shifts in semiclassical potential scattering. J. Lond. Math. Soc. (2) 91 (2015), no. 1, 159–179.MathSciNetCrossRefGoogle Scholar
  9. [GRH]
    J. Gell-Redman; A. Hassell. The distribution of phase shifts for semiclassical potentials with polynomial decay. arXiv:1509.03468
  10. [LH]
    L. Hörmander. The Analysis of Linear Partial Differential Operators I–IV. Springer-Verlag (1983, 1985).Google Scholar
  11. [Ivr1]
    V. Ivrii. The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary. (Russian) Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 25–34. English translation in Functional Analysis and Its Applications, 14:2 (1980), 98–106.Google Scholar
  12. [Ivr2]
    V. Ivrii. Semiclassical spectral asymptotics. (Proceedings of the Conference, Nantes, France, June 1991)Google Scholar
  13. [Ivr3]
    V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, SMM, 1998, xv+731.Google Scholar
  14. [Ivr4]
    V. Ivrii. Microlocal Analysis, Sharp Spectral Asymptotics and Applications.
  15. [Mel]
    R. B. Melrose. The trace of the wave group. Contemp. Math 27 (1984), 127–167.Google Scholar
  16. [Shub]
    M. Shubin. Pseudodifferential operators and spectral theory. Springer-Verlag, Berlin, 1987.Google Scholar
  17. [H-SoYa]
    A. V. Sobolev; D. R. Yafaev. Phase analysis in the problem of scattering by a radial potential. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 147:155–178, 206, 1985.Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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