TY - JOUR

T1 - Zero measure spectrum for the almost Mathieu operator

AU - Last, Y.

PY - 1994/8

Y1 - 1994/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33847667155&partnerID=8YFLogxK

U2 - 10.1007/BF02101708

DO - 10.1007/BF02101708

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AN - SCOPUS:33847667155

SN - 0010-3616

VL - 164

SP - 421

EP - 432

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -