We prove that one-dimensional Schrödinger operators with even almost periodic potential have no point spectrum for a denseG δ in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than 2 and a denseG δ in θ even if the frequency is an irrational with good Diophantine properties.
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Avron, J., Simon, B.: Almost periodic Schrödinger operators, II. The integrated density of states. Duke Math. J.50, 369–391 (1983)
del Rio, R., Jitomirskaya, S., Makarov, N., Simon, B.: Singular continuous spectrum is generic. Bull. AMS, to appear
del Rio, R., Makarov, N., Simon, B.: Operators with singular continuous spectrum, II. Rank one operators. Commun. Math. Phys., to appear
Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one-dimensional quasiperiodic Schrödinger operators. Commun. Math. Phys.132, 5–25 (1990)
Gordon, A.: On exceptional value of the boundary phase for the Schrödinger equation of a half-line. Russ. Math. Surv.47, 260–261 (1992)
Gordon, A.: Pure point spectrum under 1-paramater perturbations and instability of Anderson localization. Commun. Math. Phys., to appear
Herman, M.: Une methode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractere local d'un theoreme d'Arnold et de Moser sur le tore en dimension 2. Commun. Math. Helv.58, 453–502 (1983)
Jitomirskaya, S.: Anderson localization for the almost Mathieu equation: A nonperturbative proof, Commun. Math. Phys., to appear
Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys.164, 421–432 (1994)
Simon, B: Operators with singular continuous spectrum. I. General operators. Ann Math., to appear
Sinai, Ya.: Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential. J. Stat. Phys.46, 861–909 (1987)
Sorets, E., Spencer, T.: Positive Lyapunov exponents for Schrödinger operators with quasiperiodic potentials. Commun. Math. Phys.142, 543–566 (1991)
This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material.
Communicated by T. Spencer
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Jitomirskaya, S., Simon, B. Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators. Commun.Math. Phys. 165, 201–205 (1994). https://doi.org/10.1007/BF02099743
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