Abstract
Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl 2(ℤ), (l Y,λ Ψ)(n) = Ψ(l+1)+(l−)+λ cos(ωn+α) Ψ(n) have a purely absolutely continuous spectrum for almost all a provided that ω is a good irrational and λ is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.
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References
M. Ya. Azbel',Sov. Phys. -JETP 19:634 (1964); S. Aubry and G. André,Ann. Israel Phys. Soc. 3:133 (1980).
J. Bellissard and B. Simon,J. Funct. Anal. 48:408 (1982).
J. Avron and B. Simon,Duke Math. J. 50:369 (1983); L. A. Pastur and A. L. Figotin,J. Math. Phys. 25:774 (1984).
F. Delyon,J. Phys. A 20:L21 (1987).
E. I. Dinaburg and Ya. Sinai,Funct. Anal. Appl. 9:279 (1975); E. D. Belokolos,Theor. Math. Phys. 25:276 (1975); Ya. Sinai,Funct. Anal. Appl. 19:42 (1985); R. A. Johnson and J. Moser,Commun. Math. Phys. 84:403 (1982); H. Russmann,Ann. N. Y. Acad. Sci. 357:90 (1980).
J. Bellissard, R. Lima, and D. Testard,Commun. Math. Phys. 88:207 (1983).
Ya. Sinai,J. Stat. Phys. 46(5/6):861 (1987).
J. Fröhlich, T. Spencer, and P. Wittwer, Localization for a class of one dimensional quasiperiodic Schrödinger operators, Preprint.
I. M. Gelfand and N. Ya. Vilenkin,Generalized Functions, Vol. 4 (Academic Press, 1964).
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Chulaevsky, V., Delyon, F. Purely absolutely continuous spectrum for almost Mathieu operators. J Stat Phys 55, 1279–1284 (1989). https://doi.org/10.1007/BF01041087
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DOI: https://doi.org/10.1007/BF01041087