Scars on quantum networks ignore the lyapunov exponent

Physical Review Letters

Volumn 90, Issue 23, 2003, Pages

  Schanz, Holger ^a   Kottos, Tsampikos ^a  

^a UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

CHAOS THEORY; EIGENVALUES AND EIGENFUNCTIONS; LYAPUNOV METHODS; PROBABILITY DISTRIBUTIONS;

LYAPUNOV EXPONENT; QUANTUM NETWORKS; SCARS;

QUANTUM THEORY;

EID: 0041810591     PISSN: 00319007     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (42)

1. E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984).
2. L. Kaplan and E.J. Heller, Ann. Phys. (N.Y.) 264, 171 (1998)
3. L. Kaplan, Phys. Rev. Lett. 80, 2582 (1998).
4. E. B. Bogomolny, Physica (Amsterdam) 31D, 169 (1988)
5. R. L. Waterland et al., Phys. Rev. Lett. 61, 2733 (1988)
6. D. Wintgen and A. Hönig, Phys. Rev. Lett. 63, 1467 (1989)
7. B. Eckhardt, G. Hose, and E. Pollack, Phys. Rev. A 39, 3776 (1989)
8. M. V. Berry, Proc. R. Soc. London A 423, 219 (1989)
9. S. Tomsovic and E. J. Heller, Phys. Rev. Lett. 70, 1405 (1993)
10. D. Klakow and U. Smilansky, J. Phys. A 29, 3213 (1996)
11. F. Faure, S. Nonnenmacher, and S. De Bievre, e-print nlin.CD/0207060.
12. O. Agam and S. Fishman, Phys. Rev. Lett. 73, 806 (1994).
13. S. Fishman, B. Georgeot, and R. E. Prange, J. J. Phys. A 29, 919 (1996).
14. L. Kaplan, Phys. Rev. E 64, 036225 (2001).
15. S. Sridhar, Phys. Rev. Lett. 67, 785 (1991)
16. J. Stein and H. J. Stöckmann, Phys. Rev. Lett. 68, 2867 (1992).
17. P.B. Wilkinson et al., Nature (London) 380, 608 (1996).
18. J. P. Bird, R. Akis, and D. K. Ferry, Phys. Scr. T90, 50 (2001).
19. A. Kudrolli, M. C. Abraham, and J. P. Gollub, Phys. Rev. E 63, 026208 (2001).
20. C. Gmachl et al., Opt. Lett. 27, 824 (2002).
21. M. L. Mehta, Random Matrices (Academic Press, New York, 1991).
22. P. Kuchment, Waves Random Media 12, R1 (2002)
23. P. Kuchment, Special Issue on Quantum Graphs, edited by P. Kuchment [Waves Random Media (to be published)].
24. Proceedings of the 33rd ACM Annual Symposium on the Theory of Computing (ACM Press, New York, 2001).
25. T. Kottos and U. Smilansky, Phys. Rev. Lett. 79, 4794 (1971).
26. T. Kottos and U. Smilansky, Ann. Phys. (N.Y.) 274, 76 (1999)
27. T. Kottos and U. Smilansky, Phys. Rev. Lett. 85, 968 (2000)
28. T. Kottos and U. Smilansky, J. Phys. A 36, 3501 (2003)
29. H. Schanz and U. Smilansky, Phys. Rev. Lett. 84, 1427 (2000)
30. J. Desbois, J. Phys. A 33, L63 (2000)
31. G. Tanner, J. Phys. A 33, 3567 (2000)
32. G. Tanner, J. Phys. A 34, 8485 (2001)
33. F. Barra and P. Gaspard, J. Stat. Phys. 101, 283 (2000)
34. G. Berkolaiko and J. P. Keating, J. Phys. A 32, 7827 (1999)
35. P. Pakonski, K. Zyckowski, and M. Kus, J. Phys. A 34, 9303 (2001)
36. L. Hufnagel, R. Ketzmerick, and M. Weiss, Europhys. Lett. 54, 703 (2001)
37. G. Berkolaiko, H. Schanz, and R. S. Whitney, Phys. Rev. Lett. 88, 104101 (2002)
38. S. Gnutzmann et al., Phys. Rev. E 67, 046225 (2003).
39. G. Berkolaiko, E.B. Bogomolny, and J. P. Keating, J. Phys. A 34, 335 (2001).
40. Apparently this expectation is also supported by other scar theories which up to now have not been applied to graphs. For example, Agam and Fishman [4] predicted the positions of visible scars by a PO sum in which the orbits are weighed according to their stability.
41. For γ = 0 the scarred states satisfy the secular equation for graphs [14]. We have checked that our results remain valid in this special case.
42. Strictly speaking, δ gives only a lower bound for I, I, ≥ 1/6 - δ/3, such that the step at I = 1/6 is slightly smoothed. This is beyond the resolution of Fig. 1(b).