Zero measure spectrum for the almost Mathieu operator

Y. Last*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

109 Scopus citations

Abstract

We study the almost Mathieu operator: (Hα, λ, θu)(n)=u(n+1)+u(n-1)+λ cos (2παn+θ)u(n), on l2(Z), and show that for all λ,θ, and (Lebesgue) a.e. α, the Lebesgue measure of its spectrum is precisely |4-2|λ{norm of matrix}. In particular, for |λ|=2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational α's (and |λ|=2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.

Original languageAmerican English
Pages (from-to)421-432
Number of pages12
JournalCommunications in Mathematical Physics
Volume164
Issue number2
DOIs
StatePublished - Aug 1994
Externally publishedYes

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