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Purely Singular Continuous Spectrum for CMV Operators Generated by Subshifts

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Abstract

We prove uniform absence of point spectrum for CMV operators corresponding to the period doubling subshift. We also prove almost sure absence of point spectrum for CMV operators corresponding to a class of Sturmian subshifts. Lastly, we prove almost sure absence of point spectrum for CMV operators corresponding to some subshifts generated by a coding of a rotation.

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References

  1. Avila, A., Damanik, D., Zhang, Z.: Singular density of states measure for subshift and quasi-periodic Schrödinger operators. arXiv:1304.0519 (2013)

  2. Bellisard, J.: Spectral properties of Schrödinger’s operator with a Thue-Morse potential. In: Waldshmidt, P., Luck, M., Moussa, J.M. (eds.) From Number Theory to Physics. Springer Proceedings in Physics, vol. 47. Springer, Berlin (1990)

  3. Bellisard, J., Bovier, A., Ghez, J.-M.: Spectral properties of a tight-binding Hamiltonian with period doubling potential. Commun. Math. Phys. 135, 379–399 (1991)

    Article  ADS  Google Scholar 

  4. Bellissard, J., Iochum, B., Scoppola, E., Testard, D.: Spectral properties of one-dimensional quasicrystals. Commun. Math. Phys. 125, 527–543 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Cantero, M.-J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cantero, M.-J., Grünbaum, F.A., Moral, L., Velázquez, L.: Matrix-valued Szegö polynomials and quantum random walks. Commun. Pure Appl. Math. LXIII, 0464–0507 (2010)

    Google Scholar 

  7. Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer, Berlin (2008)

    Google Scholar 

  8. Dahl, J.: The spectrum of the off-diagonal Fibonacci operator. Ph.D. thesis, Rice University (2011)

  9. Damanik, D., Embree, M., Gorodetski, A.: Spectral properties of Schrödinger operators arising in the study of quasicrystals. arXiv:1210.5753 (2012)

  10. Damanik, D.: Gordon-type arguments in the spectral theory of one-dimensional quasicrystals. In: Baake, M., Moody, R.V. (eds.) Directions in Mathematical Quasicrystals. CRM Monograph Series, vol. 13. American Mathematical Society, Providence (2000)

  11. Damanik, D.: Strictly ergodic subshifts and associated operators. In: Gesztesy, F., Deift, P., Galvez, C., Perry, P., Schlag, W. (eds.) Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proceedings of Symposia in Pure Mathematics, vol. 76.2. American Mathematical Society, Providence (2007)

  12. Damanik, D., Killip, R., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals, III. \(\alpha \)-continuity. Commun. Math. Phys. 212, 191–204 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Damanik, D.: Uniform singular continuous spectrum for the period doubling Hamiltonian. Ann. Henri Poincaré 2, 101–108 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Damanik, D., Lenz, D.: Uniform Szegő cocycles over strictly ergodic subshifts. J. Approx. Theory 144, 133–138 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Damanik, D., Gorodetski, A.: Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian. Commun. Math. Phys. 305, 221–277 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Damanik, D., Munger, P., Yessen, W.: Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, I. The essential support of the measure. J. Approx. Theory 173, 56–88 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Damanik, David, Munger, Paul, Yessen, William: Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, II. Applications. J. Stat. Phys. 153(2), 339–362 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  18. Gesztesy, F., Zinchenko, M.: Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. J. Approx. Theory 139, 172–213 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gordon, A.: The point spectrum of the one-dimensional Schrödinger operator. Uspekhi Mat. Nauk 31(4(190)), 257–258 (1976)

    MATH  Google Scholar 

  20. Kaminaga, M.: Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential. Forum Math. 8, 63–69 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ong, D.C.: Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum. Discrete and Continuous Dynamical Systems. Proceedings of the 9th AIMS Conference Special Issue, pp. 605–609 (2013)

  22. Ong, D.C.: Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle. J. Math. Anal. Appl. 394(2), 633–644 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Peyriére, J.: Trace maps. In: Axel, F., Gratias, D. (eds.) Beyond Quasicrystals. Springer, Berlin (1995)

    Google Scholar 

  24. Simon, B.: Orthogonal Polynomaials on the Unit Circle. American Mathematical Society, Providence (2004)

    Google Scholar 

  25. Sűto, A.: The spectrum of a quasiperiodic Schrödinger operator. Commun. Math. Phys. 111, 409–415 (1987)

    Article  ADS  Google Scholar 

  26. Sűto, A.: Singular continuous spectrum on a set of zero Lebesgue measure for the Fibonacci Hamiltonian. J. Stat. Phys. 56, 525–531 (1989)

    Article  ADS  Google Scholar 

  27. Sűto, A.: Schrödinger difference equation with deterministic ergodic potentials. In: Axel, F., Gratias, D. (eds.) Beyond Quasicrystals. Springer, Berlin (1995)

    Google Scholar 

  28. Yessen, W.: Spectral analysis of tridiagonal Fibonacci Hamiltonians. J. Spectr. Theory 3(1), 101–128 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I wish to thank David Damanik and Paul Munger for helpful conversations. I also wish to thank the anonymous referees for many useful suggestions, corrections and comments. The author was supported in part by NSF Grant DMS–1067988

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Correspondence to Darren C. Ong.

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Ong, D.C. Purely Singular Continuous Spectrum for CMV Operators Generated by Subshifts. J Stat Phys 155, 763–776 (2014). https://doi.org/10.1007/s10955-014-0974-2

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  • DOI: https://doi.org/10.1007/s10955-014-0974-2

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