TY - JOUR

T1 - A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants

AU - Last, Y.

PY - 1993/1

Y1 - 1993/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33846873343&partnerID=8YFLogxK

U2 - 10.1007/BF02096752

DO - 10.1007/BF02096752

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AN - SCOPUS:33846873343

SN - 0010-3616

VL - 151

SP - 183

EP - 192

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -