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Annales Henri Poincaré

, Volume 16, Issue 11, pp 2499–2534 | Cite as

Spectral Properties of Non-Unitary Band Matrices

  • Eman Hamza
  • Alain JoyeEmail author
Article
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Abstract

We consider families of random non-unitary contraction operators defined as deformations of CMV matrices which appear naturally in the study of random quantum walks on trees or lattices. We establish several deterministic and almost sure results about the location and nature of the spectrum of such non-normal operators as a function of their parameters. We relate these results to the analysis of certain random quantum walks, the dynamics of which can be studied by means of iterates of such random non-unitary contraction operators.

Keywords

Spectral Property Polar Decomposition Quantum Walk Pure Point Random Quantum 
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References

  1. 1.
    Ahlbrecht A., Scholz V.B., Werner A.H.: Disordered quantum walks in one lattice dimension. J. Math. Phys. 52, 102201 (2011)MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Asch J., Bourget O., Joye A.: Localization properties of the Chalker–Coddington model. Ann. H. Poincaré 11, 1341–1373 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bogomolny E.: Asymptotic mean density of sub-unitary ensemble. J. Phys. A 43, 335102 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Blatter G., Browne D.: Zener tunneling and localization in small conducting rings. Phys. Rev. B 37, 3856–3880 (1988)CrossRefADSGoogle Scholar
  5. 5.
    Bourget O., Howland J.S., Joye A.: Spectral analysis of unitary band matrices. Commun. Math. Phys. 234, 191–227 (2003)zbMATHMathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Chandler-Wilde S.N., Chonchaiya R., Lindner M.: On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators. Oper. Matrices 7, 739–775 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chandler-Wilde, S.N., Lindner, M.: Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices, vol. 210, no. 989. Memoirs of the American Mathematical Society, Providence (2011)Google Scholar
  8. 8.
    Chandler-Wilde S.N., Davies E.B.: Spectrum of a Feinberg–Zee random hopping matrix. J. Spectr. Theory 2, 147–179 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Clancey, K.: Seminormal Operators, LNM. vol. 742, Springer, New York (1979)Google Scholar
  10. 10.
    Davies E.B.: Spectral theory of pseudo-ergodic operators. Comm. Math. Phys. 216, 687–704 (2001)zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Davies E.B.: Spectral properties of random non-self-adjoint matrices and operators. Proc. R. Soc. Lond. A. 457, 191–206 (2001)zbMATHCrossRefADSGoogle Scholar
  12. 12.
    Davies E.B.: Non-self-adjoint differential operators. Bull. Lond. Math. Soc. 34, 513–532 (2002)zbMATHCrossRefGoogle Scholar
  13. 13.
    Davies, E.B.: Linear Operators and their Spectra. In: Cambridge Studies in Advanced Mathematics, vol. 106. Cambridge University Press, Cambridge (2007)Google Scholar
  14. 14.
    Dimassi, M., Sjöstrand, J.: Spectral asymptotics in the semi-classical limit. In: Lecture Notes Series, vol. 268. Cambridge University Press, Cambridge (1999)Google Scholar
  15. 15.
    Feinberg J., Zee A.: Non-Hermitian localization and delocalization. Phys. Rev. E. 59, 6433–6443 (1999)CrossRefADSGoogle Scholar
  16. 16.
    Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators. In: Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence (1969)Google Scholar
  17. 17.
    Goldsheid I.Y., Khoruzhenko B.A.: Distribution of eigenvalues in non-Hermitian Anderson models. Phys. Rev. Lett. 80(13), 2897 (1998)CrossRefADSGoogle Scholar
  18. 18.
    Hamza E., Joye A.: Spectral transition for random quantum walks on trees. Commun. Math. Phys. 326, 415–439 (2014)zbMATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Hamza E., Joye A., Stolz G.: Dynamical localization for unitary Anderson models. Math. Phys. Anal. Geom. 12, 381–444 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Hatano N., Nelson D.R.: Vortex pinning and non-Hermitian quantum mechanics. Phys. Rev. B. 56, 8651–8673 (1997)CrossRefADSGoogle Scholar
  21. 21.
    Joye A.: Density of states and Thouless formula for random unitary band matrices. Ann. H. Poincaré 5, 347–379 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Joye A.: Dynamical localization for d-dimensional random quantum walks. Quantum Inf. Process. Spec. Issue Quantum Walks 11, 1251–1269 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Joye, A.: Dynamical localization of random quantum walks on the lattice. In: Jensen, A. (ed) XVIIth International Congress on Mathematical Physics, Aalborg, Denmark, 6–11 Aug 2012, World Scientific, pp. 486–494 (2013)Google Scholar
  24. 24.
    Joye A., Merkli M.: Dynamical localization of quantum walks in random environments. J. Stat. Phys. 140, 1025–1053 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1982)zbMATHCrossRefGoogle Scholar
  26. 26.
    Kempe J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003)CrossRefADSGoogle Scholar
  27. 27.
    Konno, N.: Quantum walks. In: Quantum Potential Theory. Lecture Notes in Mathematics, vol. 1954, pp. 309–452 (2009)Google Scholar
  28. 28.
    Kubrusly C.S.: Spectral Theory of Operators on Hilbert Spaces. Birkhäuser, Boston (2012)CrossRefGoogle Scholar
  29. 29.
    Reed M., Simon B.: Methods of Modern Mathematical Physics, vol. 1–4. Academic Press, New York (1979)Google Scholar
  30. 30.
    Simon, B.: Orthogonal Polynomials on the Unit Circle, Parts 1 and 2. In: Proceedings of AMS Colloquium Publications, vol. 54.1, American Mathematical Society, Providence (2005)Google Scholar
  31. 31.
    Sjöstrand, J.: Spectral properties of non-self-adjoint operators. In: Proceedings of Notes of Lectures Held in Evian les Bains, (2009). arXiv:1002.4844
  32. 32.
    Nagy B., Foias C., Berkovici H., Kérchy L.: Harmonic Analysis of Operators in Hilbert Spaces. Springer, New York (2010)CrossRefGoogle Scholar
  33. 33.
    Venegas-Andraca S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11, 1015–1106 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Trefethen L.N., Embree M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. PUP, Princeton (2005)Google Scholar
  35. 35.
    Williams J.P.: Spectra of products and numerical ranges. J. Math. Anal. Appl. 17, 214–220 (1967)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Wei Y., Fyodorov Y.V.: On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values. J. Phys. A 41, 50200 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Physics, Faculty of ScienceCairo UniversityCairoEgypt
  2. 2.UJF-Grenoble 1CNRS Institut Fourier UMR 5582GrenobleFrance

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