Skip to main content
SpringerLink
Log in

Analyticity of quantum states in one-dimensional tight-binding model

  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

Analytical complexity of quantum wavefunction whose argument is extended into the complex plane provides an important information about the potentiality of manifesting complex quantum dynamics such as time-irreversibility, dissipation and so on. We examine Pade approximation and some complementary methods to investigate the complex-analytical properties of some quantum states such as impurity states, Anderson-localized states and localized states of Harper model. The impurity states can be characterized by simple poles of the Pade approximation, and the localized states of Anderson model and Harper model can be characterized by an accumulation of poles and zeros of the Pade approximated function along a critical border, which implies a natural boundary (NB). A complementary method based on shifting the expansion-center is used to confirm the existence of the NB numerically, and it is strongly suggested that the both Anderson-localized state and localized states of Harper model have NBs in the complex extension. Moreover, we discuss an interesting relationship between our research and the natural boundary problem of the potential function whose close connection to the localization problem was discovered quite recently by some mathematicians. In addition, we examine the usefulness of the Pade approximation for numerically predicting the existence of NB by means of two typical examples, lacunary power series and random power series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Interdisciplinary Applied Mathematics, Springer, 1991)

  2. H. Yamada, K.S. Ikeda, Phys. Lett. A 222, 76 (1996)

    Article  ADS  Google Scholar 

  3. H. Yamada, K.S. Ikeda, Phys. Rev. E 65, 046211 (2002)

    Article  MATH  ADS  Google Scholar 

  4. Quantum Chaos, edited by G. Casati, B.V. Chirikov (Cambridge Univ. Press, 1996)

  5. P.W. Anderson, Phys. Rev. 109, 1492 (1958)

    Article  ADS  Google Scholar 

  6. E. Abrahams, 50 Years of Anderson Localization (World Scientific Pub Co Inc., 2010)

  7. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  8. J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd edn. (Cambridge University Press, 2004)

  9. A.O. Caldeira, A.J. Leggett, Phys. Rev. A 31, 1059 (1985)

    Article  ADS  Google Scholar 

  10. A.M. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000)

  11. G. Casati, B.V. Chirikov, J. Ford, F.M. Izrailev, in Stochastic behavior of a quantum pendulum under periodic perturbation, Lecture Note in Physics (Springer-Verlag, Berlin, 1979), Vol. 93, pp. 334–352

  12. S. Adachi, M. Toda, K. Ikeda, Phys. Rev. Lett. 61, 659 (1988)

    Article  ADS  Google Scholar 

  13. P.A. Miller, S. Sarkar, Phys. Rev. E 60, 1542 (1999)

    Article  ADS  Google Scholar 

  14. M. Goda, M. Azbel, H. Yamada, Int. J. Mod. Phys. B 13, 2705 (1999)

    Article  ADS  Google Scholar 

  15. M. Goda, M. Azbel, H. Yamada, Physica B 296, 66 (2001)

    Article  ADS  Google Scholar 

  16. V.N. Kuzovkov, W. von Niessen, Eur. Phys. J. B 42, 529 (2004)

    Article  ADS  Google Scholar 

  17. A. Peres, Phys. Rev. A 30, 1610 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  18. K. Shiokawa, B.L. Hu, Phys. Rev. E 52, 2497 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  19. T. Gorin, T. Prosen, T.H. Seligman, M. Znidaric, Phys. Rep. 435, 33 (2006)

    Article  ADS  Google Scholar 

  20. Ph. Jacquod, C. Petitjean, Adv. Phys. 58, 67 (2009)

    Article  ADS  Google Scholar 

  21. H.S. Yamada, K.S. Ikeda, Phys. Rev. E 82, 060102(R) (2010)

    Article  ADS  Google Scholar 

  22. H.S. Yamada, K.S. Ikeda, Eur. Phys. J. B 85, 41 (2012)

    Article  ADS  Google Scholar 

  23. H.S. Yamada, K.S. Ikeda, Eur. Phys. J. B 85, 195 (2012)

    Article  ADS  Google Scholar 

  24. K. Ikeda, Ann. Phys. 227, 1 (1993)

    Article  ADS  Google Scholar 

  25. A.R. Kolovsky, Phys. Rev. E 50, 3569 (1994)

    Article  ADS  Google Scholar 

  26. P.G. Harper, Proc. Phys. Soc. London Sect. A 68, 874 (1955)

    Article  MATH  ADS  Google Scholar 

  27. Y. Last, Almost everything about the almost Mathieu operator I, in XIth International Congress of Mathematical Physics, edited by D. Iagolnitzer (International Press Inc., 1995), pp. 366–372

  28. S.Y. Jitomirskaya, Almost everything about the almost Mathieu operator II, in XIth International Congress of Mathematical Physics, edited by D. Iagolnitzer (International Press Inc., 1995), pp. 373–382

  29. M. Wilkinson, E.J. Austin, Phys. Rev. B 50, 1420 (1994)

    Article  ADS  Google Scholar 

  30. H. Hiramoto, S. Abe, J. Phys. Soc. Jpn 57, 230 (1988)

    Article  ADS  Google Scholar 

  31. S. Aubry, G. Andre, Ann. Isr. Phys. Soc. 3, 133 (1980)

    MathSciNet  Google Scholar 

  32. S. Ostlund, R. Pandit, Phys. Rev. B 29, 1394 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  33. J.A. Ketoja, I.I. Satija, Phys. Rev. Lett. 75, 2762 (1995)

    Article  ADS  Google Scholar 

  34. I.I. Satija, B. Sundaram, J.A. Ketoja, Phys. Rev. E 60, 453 (1999)

    Article  ADS  Google Scholar 

  35. A. Jazaeri, I.I. Satija, Phys. Rev. E 63, 036222 (2001)

    Article  ADS  Google Scholar 

  36. K. Ishii, Prog. Theor. Phys. Suppl. 53, 77 (1973)

    Article  ADS  Google Scholar 

  37. L.M. Lifshiz, S.A. Gredeskul, L.A. Pastur, Introduction to the theory of Disordered Systems (Wiley, New York, 1988)

  38. P. Stollmann, Caught by Disorder: Bound States in Random Media (Birkhauser, 2001)

  39. H. Aoki, J. Phys. C 16, L205 (1983)

    Article  ADS  Google Scholar 

  40. C.M. Soukoulis, E.N. Economou, Phys. Rev. Lett. 52, 565 (1984)

    Article  ADS  Google Scholar 

  41. L. Pietronero, A.P. Siebesma, E. Tosatti, M. Zannetti, Phys. Rev. B 36, 5635 (1987)

    Article  ADS  Google Scholar 

  42. A.P. Siebesma, L. Pietronero, Europhys. Lett. 4, 597 (1987)

    Article  ADS  Google Scholar 

  43. M. Schreiber, Physica A 167, 188 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  44. M.V. Berry, J. Phys. A 29, 6617 (1996)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  45. D. Wojcik, I. Bialynicki-Birula, K. Zyczkowski, Phys. Rev. Lett. 85, 5022 (2000)

    Article  ADS  Google Scholar 

  46. G.A. Baker, J.L. Gammel, The Pade approximation in Theoretical Physics (Academic Press, New York, 1970)

  47. G.A. Baker Jr., Essentials of Pade approximations (Academic Press, 1975)

  48. G.A. Baker Jr., P. Graves-Morris, Pade approximations, 2nd edn. (Cambridge University Press, 1996)

  49. H.S. Yamada, K.S. Ikeda, arXiv:1308.4453v2 [math-ph] (2014)

  50. H.S. Yamada, K.S. Ikeda, Bussei Kenkyu 2, 023102 (2013) (in Japanese)

    Google Scholar 

  51. J. Breuer, B. Simon, Adv. Math. 226, 4902 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. O. Costin, M. Huang, arXiv:0810.3027 [math.CV] (2008)

  53. L.V. Ahlfors, Complex analysis (McGraw-Hill, New York, 1966)

  54. T.W. Korner, Fourier analysis (Cambridge University Press, 1988)

  55. T.W. Korner, Exercises for Fourier Analysis (Cambridge University Press, 1993)

  56. R. Remmert, Classical Topics in Complex Function Theory (Springer, New York, 2010)

  57. A. Berretti, L. Chierchia, Nonlinearity 3, 39 (1990)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  58. A. Berretti, A. Celletti, L. Chierchia, C. Falcolini, J. Stat. Phys. 66, 1613 (1992)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  59. C. Falcolini, R. Llave, J. Stat. Phys. 67, 645 (1992)

    Article  MATH  ADS  Google Scholar 

  60. R. Llave, S. Tompaidis, Physica D 71, 55 (1994)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  61. A. Berretti, S. Marmi, Chaos Solitons Fractals 5, 257 (1995)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  62. A. Berretti, C. Falcolini, G. Gentile, Phys. Rev. E 64, 015202 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  63. C. Brezinski, History of Continued Fractions and Pade Approximants (Springer Series in Computational Mathematic, 1991)

  64. J. Nuttall, J. Math. Anal. Appl. 31, 147 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  65. Ch. Pommerenke, J. Math. Anal. Appl. 41, 775 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  66. H. Stahl, J. Comp. Appl. Math. 86, 287 (1997)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  67. H. Stahl, J. Comp. Appl. Math. 99, 511 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  68. A. Shudo, K.S. Ikeda, Phys. Rev. Lett. 109, 154102 (2012)

    Article  ADS  Google Scholar 

  69. A.J. Guttmann, I.G. Enting, Phys. Rev. Lett. 76, 344 (1996)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  70. B. Nickel, J. Phys. A 32, 3889 (1999)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  71. W.P. Orrick, B.G. Nickel, A.J. Guttmann, J.H.H. Perk, Phys. Rev. Lett. 86, 4120 (2001)

    Article  ADS  Google Scholar 

  72. Y. Chan, A.J. Guttmann, B.G. Nickel, J.H.H. Perk, J. Stat. Phys. 145, 549 (2011)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  73. P. Gonnet, R. Pachon, L.N. Trefethen, Electron. Trans. Numer. Anal. 38, 146 (2011)

    MathSciNet  MATH  Google Scholar 

  74. P. Erdos, P. Turan, Ann. Math. 51, 105 (1950)

    Article  MathSciNet  Google Scholar 

  75. F. Amoroso, M. Mignotte, Ann. Inst. Fourier 46, 1275 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  76. A. Odlyzko, B. Poonen, Enseign. Math. 39, 317 (1993)

    MathSciNet  MATH  Google Scholar 

  77. B. Simon, Orthogonal Polynomials on the Unit Circle, part 1: Classical Theory (American Mathematical Society, 2004)

  78. B. Simon, Orthogonal Polynomials on the Unit Circle, part 2: Spectral Theory (American Mathematical Society, 2004)

  79. B. Simon, Szego’s Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials, (Princeton University Press, 2010)

  80. N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77, 570 (1996)

    Article  ADS  Google Scholar 

  81. N. Mott, Phil. Mag. 22, 7 (1970)

    Article  ADS  Google Scholar 

  82. Ya. Goldsheid, S. Molchanov, L. Pastur, Funct. Anal. Appl. 11, 1 (1977)

    Article  Google Scholar 

  83. S. Kotani, in Proceedings of the Kyoto Stoch. Conf., 1982

  84. B. Simon, Commun. Math. Phys. 89, 227 (1983)

    Article  MATH  ADS  Google Scholar 

  85. S. Kotani, Contemp. Math. 50, 277 (1986)

    Article  MathSciNet  Google Scholar 

  86. D. Damanik, R. Killip, Acta Math. 193, 31 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  87. H.S. Yamada, K.S. Ikeda, in preparation

  88. J.-P. Kahane, in Some Random Series of Functions, Cambridge Studies in Advanced Mathematics, 2nd edn. (Cambridge University Press, Cambridge, 1985), Vol. 5

  89. R. Carmona, A. Klein, F. Martinelli, Commun. Math. Phys. 108, 41 (1987)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  90. J. Bourgain, C. Kenig, Invent. Math. 161, 389 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yamada Physics Research Laboratory, Aoyama 5-7-14-205, Niigata, 950-2002, Japan

    Hiroaki S. Yamada

  2. Department of Physics, Ritsumeikan University, Noji-higashi 1-1-1, Kusatsu, 525, Japan

    Kensuke S. Ikeda

Authors
  1. Hiroaki S. Yamada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kensuke S. Ikeda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroaki S. Yamada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yamada, H.S., Ikeda, K.S. Analyticity of quantum states in one-dimensional tight-binding model. Eur. Phys. J. B 87, 208 (2014). https://doi.org/10.1140/epjb/e2014-50210-6

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2014-50210-6

Keywords

Search

Navigation