, Volume 153, Issue 3, pp 559-577

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

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Abstract

We consider the class of quasi-periodic self-adjoint operatorsĤ(x)) = \(\hat D(x) + \hat V(x)\) ,xS 1=ℝ1/ℤ1, on a multi-dimensional lattice ℤ v , 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 ∈ ℝ v and |V(n, x)| ≤ εe r|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.xS 1 the operatorĤ(x) has pure point spectrum. All its eigenfunctions belong tol 1(ℤ v ).

Communicated by Ya. G. Sinai