Communications in Mathematical Physics

, Volume 153, Issue 3, pp 559–577

Methods of KAM-theory for long-range quasi-periodic operators on ℤv. Pure point spectrum

  • V. A. Chulaevsky
  • E. I. Dinaburg

DOI: 10.1007/BF02096953

Cite this article as:
Chulaevsky, V.A. & Dinaburg, E.I. Commun.Math. Phys. (1993) 153: 559. doi:10.1007/BF02096953


We consider the class of quasi-periodic self-adjoint operatorsĤ(x)) =\(\hat D(x) + \hat V(x)\),xS1=ℝ1/ℤ1, on a multi-dimensional lattice ℤv, with the matrix elements
$$\hat D_{mn} (x) = \delta _{mn} D(x + n\omega ), \hat V_{mn} (x) = V(m - n, x + n\omega )$$
, whereD(x+1) =D(x), V(n, s+1) =V(n, x), ω ∈ ℝv and |V(n, x)| ≤ εer|n|,r > 0. We prove that, if ε is small enough,V(n,·) andD(·) satisfy some conditions of smoothness, andD(·) is non-degenerate, then for a.e. ω and for a.e.xS1 the operatorĤ(x) has pure point spectrum. All its eigenfunctions belong tol1(ℤv).

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • V. A. Chulaevsky
    • 1
  • E. I. Dinaburg
    • 3
  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Institute of Mathematical Problems in BiologyRussian Academy of SciencesPushchinoRussia
  3. 3.O. Yu. Schmidt Institute of Earth PhysicsRussian Academy of SciencesMoscowRussia