Skip to main content
SpringerLink
Log in

Almost-Hermitian Random Matrices: Applications to the Theory of Quantum Chaotic Scattering and Beyond

  • Chapter
Supersymmetry and Trace Formulae

Part of the book series: NATO ASI Series ((NSSB,volume 370))

Abstract

As is well-known, statistics of highly excited bound states of closed quantum chaotic systems of quite different microscopic nature is universal. Namely, it turns out to be independent of the microscopic details when sampled on the energy intervals large in comparison with the mean level separation, but smaller than the energy scale related by the Heisenberg uncertainty principle to the relaxation time necessary for the classically chaotic system to reach equilibrium in the phase space [1]. Moreover, the spectral correlation functions turn out to be exactly those which are provided by the theory of large random matrices on the local scale determined by the typical separation between neighboring eigenvalues situated around a point X, with brackets standing for the statistical averaging [2]. Microscopic justifications of the use of random matrices for describing the universal properties of quantum chaotic systems have been provided recently by several groups, based both on traditional semiclassical periodic orbit expansions [3, 4] and on advanced field-theoretical methods [5, 6]. These facts make the theory of random Hermitian matrices a powerful and versatile tool of research in different branches of modern theoretical physics, see e.g. [2, 7].

*

on leave frorn: Petersburg Nuclear ~hysica Institute, Gatchina 188350, Ruariia

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. L. Altshuler and B. D. Simon in: Mesoscopic Quantum Physics ed. by E. Akkermans et al, Les Houches Summer School, Session LXI 1994, edited E. Akkermans et al., Elsever Science.

    Google Scholar 

  2. O. Bohigas, in Chaos and Quantum Physics. Proceedings of the Les-Houches Summer School. Session LII, ed. by M. J. Giannoni et.al (North Holland, Amsterdam, 1991), p.91

    Google Scholar 

  3. M. Berry, Proc. R. Soc.London, Ser. A 400, 229 (1985

    Article  ADS  MATH  Google Scholar 

  4. E. Bogomolny and J. Keating, Phys. Rev. Lett 77, 1472 (1996)

    Article  ADS  MATH  Google Scholar 

  5. B. A. Muzykantsky and D. E. Khmelnisky JETP Lett. 62, 76 (1995)

    ADS  Google Scholar 

  6. A. Andreev, O. Agam, B. Altshuler and B. Simons Phys. Rev. Lett., 76, 3947 (1996)

    Article  ADS  Google Scholar 

  7. T. Guhr, A. Müller-Groelling and H. A. Weidenmüller, to appear in Rev.Mod.Phys.

    Google Scholar 

  8. V. V. Sokolov and V. G. Zelevinsky Phys.Lett.B 202, 10 (1988); Nucl.Phys.A 504, 562 (1989); F. Haake, F. Izrailev, N. Lehmann, D. Saher, and H.-J. Sommers, Z. Phys. B 88, 359 (1992); N. Lehmann, D. Saher, V. V. Sokolov, and H.-J. Sommers, Nucl. Phys.A 582, 223 (1995); M. Müller, F.-M. Dittes, W. Iskra, and I. Rotter, Phys.Rev.E 52, 5961 (1995).

    Article  ADS  Google Scholar 

  9. Fyodorov Y V and Sommers H-J Pis’ma ZhETF v.63, 970 (1996); [JETP Letters v.63, 1026 (1996)]

    ADS  Google Scholar 

  10. T. Gorin, F.-M. Dittes, M.Müller, I. Rotter and T. H. Seligman, Phys. Rev. E, v.56, 2481 (1997)

    Article  ADS  Google Scholar 

  11. G. Hackenbroich and J. Nöckel Europh. Lett. v.39, 371 (1997)

    Article  ADS  Google Scholar 

  12. J. Main and G. Wunner J. Phys. B: At. Mol, 27, 1994 (1994); B.Gremaud and D. Delande Europh.Lett. v.40, p.363 (1997)

    Google Scholar 

  13. R. Blumel, Phys. Rev. E, 54, 5420 (1996)

    Article  ADS  Google Scholar 

  14. V. A. Mandelshtam and H. S. Taylor, J. Chem. Soc. Faraday. Trans., 93, 847 (1997) and Phys. Rev. Lett., 78, 3274 (1997)

    Article  Google Scholar 

  15. U. Smilansky in Chaos and Quantum Physics. Proceedings of the Les-Houches Summer School. Session LII, ed. by M. J. Giannoni et.al (North Holland, Amsterdam, 1991), p.372

    Google Scholar 

  16. Y. V. Fyodorov and H.-J. Sommers, J. Math. Phys., 38, 1918 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. J. J. M. Verbaarschot, H. A. Weidenmüller, M. R. Zirnbauer, Phys.Rep. v.129, 367 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  18. C. Mahaux and H. A. Weidenmiiller, Shell Model Approach in Nuclear Reactions (North Holland, Amsterdam), 1969

    Google Scholar 

  19. I.Yu. Kobzarev, N. N. Nikolaev and L. B. Okun, Yad. Phys. 10, 864 (1969) [in Russian]

    Google Scholar 

  20. M. S. Livsic Operators, Oscillations, Waves: Open Systems, Amer. Math. Soc.Trans. v.34 (Am.Math.Soc., Providence, RI, 1973)

    Google Scholar 

  21. K. B. Efetov Supersymmetry in Disorder and Chaos (Cambridge University Press,1996)

    Google Scholar 

  22. Y. V. Fyodorov in “Mesosocopic Quantum Physics”, Les Houches Summer School, Session LXI, 1994, edited E. Akkermans et al., Elsever Science, p.493

    Google Scholar 

  23. Y. V. Fyodorov and Y. Alhassid, Phys. Rev. A, 58, 3375 (1998)

    Article  ADS  Google Scholar 

  24. D. V. Savin and V. V. Sokolov, Phys. Rev. E v.56, R4911 (1997)

    Article  ADS  Google Scholar 

  25. E. Gudowska-Nowak, G. Papp and J. Brickmann Chem. Physics v.220, 125 (1997)

    Article  ADS  Google Scholar 

  26. R. Grobe, F. Haake, and H.-J. Sommers, Phys. Rev. Lett. 61, 1899 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  27. F. Haake Quantum Signature of Chaos (Berlin, Springer, 1991)

    Google Scholar 

  28. L. E. Reichl, Z. Y. Chen and M. Millonas Phys. Rev. Lett. v.63, 2013 (1989)

    Article  ADS  Google Scholar 

  29. H.-J. Sommers, A. Crisanti, H. Sompolinsky, and Y. Stein, Phys. Rev. Lett. 60, 1895 (1988); H. Sompolinsky, A. Crisanti and H.-J. Sommers Phys. Rev. Lett. 61 259, 1988; B. Doyon, B. Cessac, M. Quoy and M. Samuelidis Int. J. Bifurc. Chaos 3 279 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  30. N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996); Phys. Rev. B v.56 (1997), 8651

    Article  ADS  Google Scholar 

  31. P. W. Brouwer, P. G. Silvestrov and C. W. J. Beenakker Phys. Rev. B v.56 (1997), 4333; R. A. Janik et al., e-preprint cond-mat/9705098; B. A. Khoruzhenko and I. Goldscheid Phys. Rev. Lett. 80, 2897 (1998)

    Article  ADS  Google Scholar 

  32. K. B. Efetov, Phys. Rev. Lett. 79, 491 (1997)

    Article  ADS  Google Scholar 

  33. J. Miller and J. Wang, Phys. Rev. Lett. 76, 1461 (1996); J. Chalker and J. Wang, Phys. Rev. Lett. 79, 1797 (1997)

    Article  ADS  Google Scholar 

  34. M. A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996); R. A.Janik et al., Phys. Rev. Lett. 77, 4876 (1996); M. A. Halasz, A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D, v.56, 5140 (1997); M. A. Halasz, J. C.Osborn and J. J. M. Verbaarschot Phys. Rev. D, 56, 7059 (1997).

    Article  ADS  Google Scholar 

  35. T. Akuzawa and M. Wadati, J. Phys. Soc. Jpn. 65, 1583 (1996).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. M. V. Feigelman and M. A. Skvortsov, Nucl. Phys. B 506, 665 (1997); A. Khare and K. Ray, Phys. Lett. A 230, 139 (1997).

    Article  ADS  Google Scholar 

  37. B. A. Khoruzhenko, J. Phys. A 29, L165 (1996).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Y. V. Fyodorov, B. Khoruzhenko and H.-J. Sommers, Physics Letters A 226, 46 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Y. V. Fyodorov, B. Khoruzhenko and H.-J. Sommers, Phys.Rev.Lett. v. 79, 557 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. G. Oas, Phys. Rev. E 55, 205 (1997)

    Article  ADS  Google Scholar 

  41. R. A. Janik, M. Nowak. G. Papp and I. Zahed Nucl.Phys. B 501, 603 (1997); J. Feinberg and A. Zee, Nucl.Phys. B. 501, 643 (1997) and Nucl.Phys.B. 504, 579 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. M. Kus, F. Haake, D. Zaitsev and A. Huckleberry, J. Phys. A: Math. Gen. 30, 8635 (1997).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. J. Ginibre, J. Math. Phys. 6, 440 (1965).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. M. L. Mehta, Random Matrices (Academic Press Inc., N. Y., 1990)

    Google Scholar 

  45. V. Girko, Theor. Prob. Appl. 30, 677 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  46. N. Lehmann and H.-J. Sommers, Phys.Rev.Lett. 67, 941 (1991).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. P. J. Forrester, Phys. Lett. A 169, 21 (1992); J. Phys. A: Math. Gen. 26, 1179 (1993).

    Article  MathSciNet  ADS  Google Scholar 

  48. A. Edelman, J. Multivariate Anal. 60, 203 (1997); A.Edelman, E.Kostlan and M.Shub J.Am.Math.Soc. v.7, 247 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  49. Z. D. Bai, Ann. Prob. 25, 494 (1997).

    Article  MATH  Google Scholar 

  50. Y. V. Fyodorov, B. A. Khoruzhenko and H.-J. Sommers Ann.Inst.Henri Poincare: Physique Theorique, 68, 449 (1998)

    MathSciNet  MATH  Google Scholar 

  51. C. H. Lewenkopf and H. A. Weidenmüller, Ann.Phys. v.212, 53 (1991)

    Article  ADS  MATH  Google Scholar 

  52. P. Gaspard in “Quantum Chaos”, Proceedings of E. Fermi Summer School, 1991 ed. by G. Casati et al. (North Holland, Amsterdam,1991), p.307

    Google Scholar 

  53. D. Stone in Mesoscopic Quantum Physics, see [1]

    Google Scholar 

  54. R. Schinke, H.-M. Keller, M. Stumpf and A. J. Dobbyn, J. Phys. B: At. Mol., 28, 2928 (1995)

    Google Scholar 

  55. V. V. Flambaum, A. A. Gribakina and G. F. Gribakin, Phys. Rev. A v.54, 2066, (1996)

    Article  ADS  Google Scholar 

  56. P. A. Moldauer, Phys.Rev. v.157, 907 (1967); M. Simonius Phys. Lett. v.52B, 279 (1974)

    Article  ADS  Google Scholar 

  57. S. Albeverio, F. Haake, P. Kurasov, M. Kus and P. Seba, J. Math. Phys. v.37, 4888 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. H. L. Harney, F. M. Dittes and A. Müller Ann.Phys. v.220, 159 (1992)

    Article  ADS  MATH  Google Scholar 

  59. N. Lehmann, D. Savin, V. V. Sokolov and H.-J. Sommers Physica D 86 572 (1995); Y. V. Fyodorov, D. Savin and H.-J. Sommers, Phys. Rev. E, 55, 4857 (1997)

    Article  MATH  Google Scholar 

  60. R. Schinke Photodissociation Dynamics, (Cambridge University, 1993)

    Google Scholar 

  61. V. V. Sokolov and V. G. Zelevinsky Phys. Rev. C 56, 311 (1997)

    Article  ADS  Google Scholar 

  62. N. Taniguchi, A. V. Andreev and B. L. Altshuler, Europh. Lett., 29, 515 (1995)

    Article  ADS  Google Scholar 

  63. A. Pandey and M. L. Mehta, Commun. Math. Phys. v.87, 449 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. A. Altland, K. B. Efetov, S. Iida, J. Phys. A: Math.Gen v.26, 2545 (1993)

    Article  Google Scholar 

  65. A. D. Mirlin, Y. V. Fyodorov, J. Phys. A v.24, 2273 (1991); Y. V. Fyodorov, A. D. Mirlin, Phys. Rev. Lett. v.67, 2049 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. Y. V. Fyodorov and H.-J. Sommers, Z. Phys. B v.99, 123 (1995)

    Article  ADS  Google Scholar 

  67. Strictly speaking, the form of the correlation function of eigenvalue densities for sparse matrices was shown to be identical to that known for the corresponding Gaussian ensemble provided p exceeds some critical value p = p l. The “threshold” value p l is nonuniversal and depends on the form of the distribution P(Ĥ) [65]. However, direct numerical simulations, see S. Evangelou J. Stat. Phys. v.69 (1992), 361 show that actual value is 1 < p l < 2. Thus, even existence of two nonvanishing elements per row already ensure, that the corresponding statistics belongs to the Gaussian universality class. In the present paper we assume that p> p l.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. G. Hackenbroich and H. A. Weidenmüller Phys. Rev. Lett. 74 4118 (1995)

    Article  ADS  Google Scholar 

  69. F. Di Francesco, M. Gaudin, C. Itzykson, and F. Lesage Int. J. Mod. Phys. A 9, 4257 (1994).

    Article  ADS  MATH  Google Scholar 

  70. P. J. Forrester and B. Jancovici, Int. J. Mod.Phys.A 11, 941 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  71. G. Szegö, Orthogonal polynomials, 4th ed. (AMS, Providence, 1975), p. 380.

    MATH  Google Scholar 

  72. I. S. Gradshteyn, I. M. Ryzhik “Table of Integrals, Series, and Products” (Academic Press, N. Y. 1980).

    MATH  Google Scholar 

  73. Y. V. Fyodorov, M. Titov and H.-J. Sommers, Phys. Rev. E, 58, 1195 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  74. H.-J. Sommers, Y. V. Fyodorov and M. Titov, e-preprint cond-mat/9807015

    Google Scholar 

  75. Y. V. Fyodorov and B. A. Khoruzhenko, under preparation

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Physik, Universität-GH Essen, D-45117, Essen, Germany

    Yan V. Fyodorov

Authors
  1. Yan V. Fyodorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Igor V. Lerner

  2. University of Bristol and Hewlett-Packard Laboratories, Bristol, UK

    Jonathan P. Keating

  3. University of Cambridge, Cambridge, UK

    David E. Khmelnitskii

  4. L. D. Landau Institute for Theoretical Physics, Moscow, Russia

    David E. Khmelnitskii

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media New York

About this chapter

Cite this chapter

Fyodorov, Y.V. (1999). Almost-Hermitian Random Matrices: Applications to the Theory of Quantum Chaotic Scattering and Beyond. In: Lerner, I.V., Keating, J.P., Khmelnitskii, D.E. (eds) Supersymmetry and Trace Formulae. NATO ASI Series, vol 370. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4875-1_15

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-4875-1_15

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7212-7

  • Online ISBN: 978-1-4615-4875-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation