Abstract
We prove that Schrödinger operators with meromorphic potentials \({(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n}\) have purely singular continuous spectrum on the set \({\{E: L(E) < \delta{(\alpha, \theta)}\}}\), where \({\delta}\) is an explicit function and L is the Lyapunov exponent. This extends results of Jitomirskaya and Liu (Arithmetic spectral transitions for the Maryland model. CPAM, to appear) for the Maryland model and of Avila,You and Zhou (Sharp Phase transitions for the almost Mathieu operator. Preprint, 2015) for the almost Mathieu operator to the general family of meromorphic potentials.
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References
Avila, A.: Absolutely continuous spectrum for the almost Mathieu operator. Preprint arXiv:0810.2965v1 (2008)
Avila A., Jitomirskaya S.: The ten Martini problem. Ann. Math. 170, 303–342 (2009)
Avila A., Jitomirskaya S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)
Avila, A., You, J., Zhou, Q.: Sharp Phase transitions for the almost Mathieu operator. Preprint (2015)
Grempel D., Fishman S., Prange R.: Localization in an incommensurate potential: an exactly solvable model. Phys. Rev. Lett. 49(11), 833 (1982)
Gordon A.: The point spectrum of the one-dimensional Schrödinger operator. Uspehi Mat. Nauk 31, 257–258 (1976)
Ganeshan S., Kechedzhi K., Das Sarma S.: Critical integer quantum hall topology and the integrable maryland model as a topological quantum critical point. Phys. Rev. B 90, 041405 (2014)
Jitomirskaya, S.: Almost everything about the almost Mathieu operator, II. In: Proceedings of XI International Congress of Mathematical Physics, 373–382. International Press (1995)
Jitomirskaya S.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999)
Jitomirskaya S.: Ergodic Schrödinger operators (on one foot). Proc. Symp. Pure Math. 76, 613–647 (2007)
Jitomirskaya, S., Marx, C.A.: Dynamics and spectral theory of quasi-periodic Schrödinger-type operators. ETDS, to appear
Jitomirskaya, S., Liu, W.: Universal hierarchical structure of quasiperiodic eigenfunctions. arXiv:1609.08664
Jitomirskaya, S., Liu, W.: Universal reflexive-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transitions in phase. Preprint
Jitomirskaya, S., Liu, W.: Arithmetic spectral transitions for the Maryland model. CPAM (to appear)
Jitomirskaya S., Simon B.: Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators. Commun. Math. Phys. 165, 201–205 (1994)
Last, Y.: Spectral theory of Sturm–Liouville operators on infinite intervals: a review of recent developments. In: Amrein, W.O., Hinz, A.M., Pearson, D.P. (eds.) Sturm–Liouville theory, pp. 99–120. Birkhauser, Basel (2005)
Simon B., Spencer T.: Trace class perturbations and the absence of absolutely continuous spectra. Commun. Math. Phys. 125(1), 113–125 (1989)
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Communicated by J. Marklof
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Jitomirskaya, S., Yang, F. Singular Continuous Spectrum for Singular Potentials. Commun. Math. Phys. 351, 1127–1135 (2017). https://doi.org/10.1007/s00220-016-2823-4
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DOI: https://doi.org/10.1007/s00220-016-2823-4