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

Y. Last*

*Corresponding author for this work

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Abstract

We study ergodic Jacobi matrices on l2(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: (Hα, λ, θu)(n)=u(n+1)+u(n-1)+λ cos(2παn+θ)u(n), and prove the existence of a.c. spectrum for sufficiently small λ, all irrational α's, and a.e. θ. Moreover, for 0≤λ<2 and (Lebesgue) a.e. pair α, θ, we prove the explicit equality of measures: |σac|=|σ|=4 -2λ.

Original languageAmerican English
Pages (from-to)183-192
Number of pages10
JournalCommunications in Mathematical Physics
Volume151
Issue number1
DOIs
StatePublished - Jan 1993
Externally publishedYes

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