Abstract
We discuss the recent proof of Cantor spectrum for the almost Mathieu operator for all conjectured values of the parameters.
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References
S. Aubry, The new concept of transition by breaking of analyticity. Solid State Sci. 8, 264 (1978).
A. Avila and S. Jitomirskaya, The ten martini problem. Preprint (www.arXiv.org).
A. Avila and R. Krikorian, Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Preprint (www.arXiv.org). To appear in Annals of Math.
J. E. Avron, D. Osadchy, and R. Seiler, A topological look at the quantum Hall effect, Physics today, 38–42, August 2003.
J. Avron and B. Simon, Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J. 50, 369–391 (1983).
J. Avron, P. van Mouche, and B. Simon, On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys. 132, 103–118 (1990).
M. Ya. Azbel, Energy spectrum of a conduction electron in a magnetic field. Sov. Phys. JETP 19, 634–645 (1964).
J. Bellissard, B. Simon, Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48, 408–419 (1982).
J. Bourgain, S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Statist. Phys. 108, 1203–1218 (2002).
J. Bourgain, S. Jitomirskaya, Absolutely continuous spectrum for 1D quasiperiodic operators, Invent. math. 148, 453–463 (2002).
M.D. Choi, G.A. Eliott, N. Yui, Gauss polynomials and the rotation algebra. Invent. Math. 99, 225–246 (1990).
L. H. Eliasson, Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992).
S. Jitomirskaya, Metal-Insulator transition for the almost Mathieu operator. Annals of Math. 150, 1159–1175 (1999).
S. Ya. Jitomirskaya, I. V. Krasovsky, Continuity of the measure of the spectrum for discrete quasiperiodic operators. Math. Res. Lett. 9, no. 4, 413–421 (2002).
P.G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. London A 68, 874–892 (1955).
B. Helffer and J. Sjöstrand, Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum. Mm. Soc. Math. France (N.S.) 39, 1–124 (1989).
S. Kotani, Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrö dinger operators. Stochastic analysis (Katata/Kyoto, 1982), 225–247, North-Holland Math. Library, 32, North-Holland, Amsterdam, 1984.
Y. Last, Zero measure of the spectrum for the almost Mathieu operator, CMP 164, 421–432 (1994).
Y. Last, Spectral theory of Sturm-Liouville operators on infinite intervals: a review of recent developments. Preprint 2004.
P.M.H. van Mouche, The coexistence problem for the discrete Mathieu operator, Comm. Math. Phys., 122, 23–34 (1989).
R. Peierls, Zur Theorie des Diamagnetismus von Leitungselektronen. Z. Phys., 80, 763–791 (1933).
J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244, 297–309 (2004).
A. Rauh, Degeneracy of Landau levels in chrystals, Phys. Status Solidi B 65, K131–135 (1974).
B. Simon, Kotani theory for one-dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89, 227–234 (1983).
B. Simon, Schrö dinger operators in the twenty-first century, Mathematical Physics 2000, Imperial College, London, 283–288.
Ya. Sinai, Anderson localization for one-dimensional difference Schrö dinger operator with quasi-periodic potential. J. Stat. Phys. 46, 861–909 (1987).
D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den Nijs, Quantised Hall conductance in a two dimensional periodic potential, Phys. Rev. Lett. 49, 405–408 (1982).
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Avila, A., Jitomirskaya, S. (2006). Solving the Ten Martini Problem. In: Asch, J., Joye, A. (eds) Mathematical Physics of Quantum Mechanics. Lecture Notes in Physics, vol 690. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-34273-7_2
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DOI: https://doi.org/10.1007/3-540-34273-7_2
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