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An Example of Embedded Singular Continuous Spectrum for Discrete Schrödinger Operators

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Abstract

We present an example of a potential such that the corresponding discrete Schrödinger operator has singular continuous spectrum embedded in the absolutely continuous spectrum.

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References

  1. Delyon, F., Simon, B., Souillard, B.: From pure point to continuous spectrum in disordered systems. Ann. Henri Poincaré, 42, 283–309 (1985)

    MathSciNet  Google Scholar 

  2. Denisov, S. A., Kiselev, A.: Spectral properties of Schrödinger operators with decaying potentials, In: Spectral Theory and Mathematical Physics, American Mathematical Society, Providence, RI, 2007, 565–589

    Google Scholar 

  3. Grafakos, L.: Classical Fourier Analysis, Grad. Texts in Math., vol. 249, Springer-Verlag, New York, 2008

    Google Scholar 

  4. Kiselev, A.: Imbedded singular continuous spectrum for Schrödinger operators. J. Amer. Math. Soc., 18, 571–603 (2005)

    Article  MathSciNet  Google Scholar 

  5. Kiselev, A., Last, Y., Simon, B.: Modified Pröfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Comm. Math. Phys., 194, 1–45 (1998)

    Article  MathSciNet  Google Scholar 

  6. Kiselev, A., Remling, C., Simon, B.: Effective perturbation methods for one-dimensional Schrödinger operators. J. Differential Equations, 151, 290–312 (1999)

    Article  MathSciNet  Google Scholar 

  7. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math., 135, 329–367 (1999)

    Article  MathSciNet  Google Scholar 

  8. Liu, W.: Absence of singular continuous spectrum for perturbed discrete Schrödinger operators. J. Math. Anal. Appl., 472, 1420–1429 (2019)

    Article  MathSciNet  Google Scholar 

  9. Lukic, M., Ong, D. C.: Generalized Pröfer variables for perturbations of Jacobi and CMV matrices. J. Math. Anal. Appl., 444, 1490–1514 (2016)

    Article  MathSciNet  Google Scholar 

  10. Menchoff, D.: Sur les séries de fonctions orthogonales. Fund. Math., 10, 375–420 (1927)

    Article  MathSciNet  Google Scholar 

  11. Pearson, D.: Singular continuous measures in scattering theory. Comm. Math. Phys., 60, 13–36 (1978)

    Article  MathSciNet  Google Scholar 

  12. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis. Second Edition, Academic Press, New York, 1980

    Google Scholar 

  13. Remling, C.: Embedded singular continuous spectrum for one-dimensional Schröodinger operators, Trans. Amer. Math. Soc., 351, 2479–2497 (1999)

    Article  MathSciNet  Google Scholar 

  14. Simon, B.: Spectral analysis of rank one perturbations and applications, CMR Proceedings and Lecture Notes, 8, 109–149 (1995)

    Article  MathSciNet  Google Scholar 

  15. Simon, B., Stolz, G.: Operators with singular continuous spectrum. V. Sparse potentials, Proc. Amer. Math. Soc., 124, 2073–2080 (1996)

    Article  MathSciNet  Google Scholar 

  16. Stout, W. F.: Almost Sure Convergence, Academic Press, New York, 1974

    Google Scholar 

  17. Tchebotareva, O.: An example of embedded singular continuous spectrum for one-dimensional Schröodinger operators. Lett. Math. Phys., 72, 225–231 (2005)

    Article  MathSciNet  Google Scholar 

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Correspondence to Xiong Li.

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Conflict of Interest The authors declare no conflict of interest.

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Supported by National Natural Science Foundation of China (Grant No. 12371158)

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Fu, Z.Q., Li, X. An Example of Embedded Singular Continuous Spectrum for Discrete Schrödinger Operators. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-2574-7

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  • DOI: https://doi.org/10.1007/s10114-024-2574-7

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