Abstract
We study ergodic Jacobi matrices onl 2(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: (H α, λ, θ u)(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.
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Aubry, S., Andre, G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc.3, 133–164 (1980)
Avron, J., Simon, B.: Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J.50, 369–391 (1983)
Avron, J., van Mouche, P., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys.132, 103–118 (1990)
Bellissard, J., Lima, R., Testard, D.: A metal-insulator transition for the almost Mathieu model. Commun. Math. Phys.88, 207–234 (1983)
Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal.48, 408–419 (1982)
Bhatia, R.: Perturbation bounds for matrix eigenvalues. Pitman Research Notes in Mathematics Series162, Essex: Longman 1987
Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math.99, 225–246 (1909)
Chulaevsky, V., Delyon, F.: Purely absolutely continuous spectrum for almost Mathieu operators. J. Stat. Phys.55, 1279–1284 (1989)
Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer 1987
Deift, P., Simon, B.: Almost periodic Schrödinger operators. III.The absolutely continuous spectrum in one dimension. Commun. Math. Phys.90, 389–411 (1983)
Delyon, F.: Absence of localization for the almost Mathieu equation. J. Phys. A20, L21-L23 (1987)
Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys.132, 5–25 (1990)
Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat.31, 457–469 (1960)
Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, Fifth ed. Oxford: Oxford University Press 1979
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
Last, Y.: On the measure of gaps and spectra for discrete 1D Schrödinger operators. Commun. Math. Phys. (to appear)
Simon, B.: Almost periodic Schrödinger operators. A review. Adv. Appl. Math.3, 463–490 (1982)
Sinai, Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys.46, 861–909 (1987)
Thouless, D.J.: Bandwidth for a quasiperiodic tight binding model. Phys. Rev. B28, 4272–4276 (1983)
Toda, M.: Theory of nonlinear lattices. 2nd. ed. chap. 4. Berlin, Heidelberg, New York: Springer 1989
Zaanen, A.C.: An introduction to the theory of integration. Amsterdam: North-Holland 1958
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Communicated by B. Simon
Work partially supported by the US-Israel BSF
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Last, Y. A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Commun.Math. Phys. 151, 183–192 (1993). https://doi.org/10.1007/BF02096752
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DOI: https://doi.org/10.1007/BF02096752