Abstract
We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twenty-first century.
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References
Aubry, S., André, G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133–164 (1980)
Avila, A., Jitomirskaya, S.: The ten Martini problem. Ann. Math. (to appear)
Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)
Avila, A., Krikorian, R.: Quasiperiodic SL(2,ℝ) cocycles. In preparation
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)
Avron, J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc. 6, 81–85 (1982)
Avron, J., Simon, B.: Almost periodic Schrödinger operators. II: The integrated density of states. Duke Math. J. 50, 369–391 (1983)
Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48, 408–419 (1982)
Bourgain, J., Jitomirskaya, S.: Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Statist. Phys. 108, 1203–1218 (2002)
Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99, 225–246 (1990)
Damanik, D.: Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: A survey of Kotani theory and its applications. In: Spectral Theory and Mathematical Physics: a Festschrift in honor of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math., vol. 76, Part 2, pp. 539–563. Am. Math. Soc., Providence, RI (2007)
Deift, P., Simon, B.: Almost periodic Schrödinger operators. III: The absolutely continuous spectrum in one dimension. Commun. Math. Phys. 90, 389–411 (1983)
Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. 154, 155–203 (2001)
Goldstein, M., Schlag, W.: Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues. Geom. Funct. Anal. (to appear)
Gordon, A.: On the point spectrum of the one-dimensional Schrödinger operator. Usp. Math. Nauk. 31, 257–258 (1976)
Gordon, A., Jitomirskaya, S., Last, Y., Simon, B.: Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178, 169–183 (1997)
Jitomirskaya, S.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159–1175 (1999)
Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982)
Kotani, S.: Generalized Floquet theory for stationary Schrödinger operators in one dimension. Chaos Solitons Fractals 8, 1817–1854 (1997)
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)
Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164, 421–432 (1994)
Last, Y.: Spectral theory of Sturm-Liouville operators on infinite intervals: A review of recent developments. In: Sturm-Liouville Theory, pp. 99–120. Birkhäuser, Basel (2005)
Puig, J.: Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244, 297–309 (2004)
Simon, B.: Almost periodic Schrödinger operators: A review. Adv. Appl. Math. 3, 463–490 (1982)
Simon, B.: Kotani theory for one dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983)
Simon, B.: Schrödinger operators in the twenty-first century. In: Mathematical Physics 2000, pp. 283–288. Imp. Coll. Press, London (2000)
Thouless, D.: Bandwidths for a quasiperiodic tight binding model. Phys. Rev. B 28, 4272–4776 (1983)
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Avila, A., Damanik, D. Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling . Invent. math. 172, 439–453 (2008). https://doi.org/10.1007/s00222-007-0105-7
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DOI: https://doi.org/10.1007/s00222-007-0105-7