A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants

Abstract

We study ergodic Jacobi matrices on 1 2 (Z), and prove a general theorem relating their a.c. spectrum to the spectra of periodic Jacobi matrices, that are obtained by cutting finite pieces from the ergodic potential and then repeating them. We apply this theorem to the almost Mathieu operator: (H a xθ u) (n) = u(n + 1) + u(n -1) + λcos(2παn + θ)u(n), and prove the existence of a.c. spectrum for sufficiently small λ, all irrational α's, and a.e. 0. Moreover, for 0 < λ < 2 and (Lebesgue) a.e. pair α, 0, we prove the explicit equality of measures: |σ ac | = |σ| = 4 -2λ.

Continuity of the measure of the spectrum for discrete quasiperiodic operators

“…This improves some results of [16,13] and establishes the Aubry-Andre conjecture on the measure of the spectrum for all values of parameters in the noncritical case. For the critical case, λ = 2, zero measure of the spectrum is known for the full measure set of α [11,17], however extending it to all irrational α remains an open problem.…”

“…For α whose coefficients of the continued fraction expansion form an unbounded sequence, the condition γ(E, α) > 0 is redundant. The result of Theorem 1 follows in this case (by a simple generalization of the argument in [16]) from the Hölder-1/2 continuity of the spectrum established in [3]. For α's whose continued fraction coefficients form a bounded sequence (henceforth, we denote the set of such α by Ω) this continuity is not sufficient to imply Theorem 1.…”

“…That argument was improved to the Hölder-1/2 continuity (for arbitrary f ∈ C 1 ) in [3] and subsequently used in [16,17] to establish (1.2) for the almost Mathieu operator for α with unbounded continuous fraction expansion, therefore proving the Aubry-Andre conjecture for a.e. (but not all) α.…”

Schrödinger operators in the twentieth century

“…Then ͑i͒ If Ͻ2, there is always ͑i.e., for any irrational ␣͒ lots of a.c. spectrum and it is known for some ␣ and believed for all ␣ that is all there is ͑see Last, 169 Gesztesy and Simon, 92 Gordon et al, 100 Jitomirskaya; 134 the earliest results of this genre are due to Dinaburg and Sinai 67 ͒.…”

Dimensional Hausdorff Properties of Singular Continuous Spectra

“…2. While the jlj , 2 part of this conjecture may be correct (so far, the existence of the absolutely continuous spectrum [13] and the absence of the point spectrum [14] have been established rigorously), the jlj . 2 case turned out to be more delicate: The absolutely continuous spectrum is absent [15], but both pure-point and singular-continuous spectra occur, depending on arithmetical properties of both b and u [16].…”