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 -