Mathematics Spectrum January 2017 Pdf

• Published: 17 January 2017

Singular Continuous Spectrum for Singular Potentials

Communications in Mathematical Physics volume 351,pages 1127–1135 (2017)Cite this article

• 240 Accesses

• 8 Citations

• 1 Altmetric

• Metrics details

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.

Access options

Buy single article

Instant access to the full article PDF.

34,95 €

Price includes VAT (Singapore)
Tax calculation will be finalised during checkout.

References

1. 1

   Avila, A.: Absolutely continuous spectrum for the almost Mathieu operator. Preprint arXiv:0810.2965v1 (2008)

2. 2

   Avila A., Jitomirskaya S.: The ten Martini problem. Ann. Math. 170, 303–342 (2009)

   MathSciNet  Article  MATH  Google Scholar

3. 3

   Avila A., Jitomirskaya S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)

   MathSciNet  Article  MATH  Google Scholar

4. 4

   Avila, A., You, J., Zhou, Q.: Sharp Phase transitions for the almost Mathieu operator. Preprint (2015)

5. 5

   Grempel D., Fishman S., Prange R.: Localization in an incommensurate potential: an exactly solvable model. Phys. Rev. Lett. 49(11), 833 (1982)

   ADS  Article  Google Scholar

6. 6

   Gordon A.: The point spectrum of the one-dimensional Schrödinger operator. Uspehi Mat. Nauk 31, 257–258 (1976)

   MathSciNet  Google Scholar

7. 7

   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)

   ADS  Article  Google Scholar

8. 8

   Jitomirskaya, S.: Almost everything about the almost Mathieu operator, II. In: Proceedings of XI International Congress of Mathematical Physics, 373–382. International Press (1995)

9. 9

   Jitomirskaya S.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999)

   MathSciNet  Article  MATH  Google Scholar

10. 10

    Jitomirskaya S.: Ergodic Schrödinger operators (on one foot). Proc. Symp. Pure Math. 76, 613–647 (2007)

    Article  MATH  Google Scholar

11. 11

    Jitomirskaya, S., Marx, C.A.: Dynamics and spectral theory of quasi-periodic Schrödinger-type operators. ETDS, to appear

12. 12

    Jitomirskaya, S., Liu, W.: Universal hierarchical structure of quasiperiodic eigenfunctions. arXiv:1609.08664

13. 13

    Jitomirskaya, S., Liu, W.: Universal reflexive-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transitions in phase. Preprint

14. 14

    Jitomirskaya, S., Liu, W.: Arithmetic spectral transitions for the Maryland model. CPAM (to appear)

15. 15

    Jitomirskaya S., Simon B.: Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators. Commun. Math. Phys. 165, 201–205 (1994)

    ADS  Article  MATH  Google Scholar

16. 16

    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)

17. 17

    Simon B., Spencer T.: Trace class perturbations and the absence of absolutely continuous spectra. Commun. Math. Phys. 125(1), 113–125 (1989)

    ADS  MathSciNet  Article  MATH  Google Scholar

Download references

Author information

Affiliations

1. University of California, Irvine, CA, 92697, USA

   Svetlana Jitomirskaya

2. Ocean University of China, Qingdao, China

   Fan Yang

Corresponding author

Correspondence to Fan Yang.

Additional information

Communicated by J. Marklof

Rights and permissions

About this article

Cite this article

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

Download citation

• Received: 18 April 2016

• Accepted: 16 November 2016

• Published: 17 January 2017

• Issue Date: May 2017

• DOI : https://doi.org/10.1007/s00220-016-2823-4

Mathematics Spectrum January 2017 Pdf

Source: https://link.springer.com/article/10.1007/s00220-016-2823-4

Posted by: mcdanielalsorombicks.blogspot.com

Share this post