Abstract
Random matrix theory (RMT) serves as a very useful backdrop for the discussion of classical chaos and its effects on quantum mechanics. Two important suggestions began this fruitful association: the conjecture by Bohigas, Giannoni, and Schmit1 associated RMT phenomenology with quantum eigenstates of classically chaotic systems. Berry’s conjecture2 stated that quantum eigenstates of classically chaotic systems should locally look like random superpositions of plane waves of the same (local) wavevector magnitude k, producing Gaussian random fluctuations in position space. These two closely related ideas turn out to be the clay out of which the truth is molded. Much research has centered on modifications to the extreme uniformity of RMT, modifications imposed by known constraints on the quantum dynamics. In fact there is no Hamiltonian system whose eigenstates are known to hold to the rigors of true Gaussian randomness.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
O. Bohigas, M.-J. Giannoni, and C. Schmit, J. Physique Lett. 45, L–1015 (1984).
M. V. Berry, in Chaotic Behaviour of Deterministic Systems, ed. by G. Iooss, R. Helleman, and R. Stora (North-Holland 1983) p. 171.
E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984).
P. O’Connor, J. N. Gehlen, and E. J. Heller, Phys. Rev. Lett. 58, 1296 (1987).
S. Sridhar, Phys. Rev. Lett 67, 785 (1991).
J. Stein and H.-J. Stöckman, Phys. Rev. Lett. 68, 2867 (1992).
T. M. Fromhold, P. B. Wilkinson, F. W. Sheard, L. Eaves, J. Miao, and G. Edwards, Phys. Rev. Lett. 75, 1142 (1995); P. B. Wilkinson, T. M. Promhold, L. Eaves, F. W. Sheard, N. Miura, and T. Takamasu, Nature 380, 608 (1996).
D. Wintgen and A. Honig, Phys. Rev. Lett. 63, 1467 (1989).
K. Muller and D. Wintgen, J. Phys. B 27, 2693 (1994).
E. B. Bogomolny, Physica D 31, 169 (1988).
M. V. Berry, Les Houches Lecture Notes, Summer School on Chaos and Quantum Physics, M-J. Giannoni, A. Voros, and J. Zinn-Justin, eds., Elsevier Science Publishers B. V. (1991); M. V. Berry, Proc. Roy. Soc. A 243, 219 (1989).
R. Aurich and F. Steiner, Chaos, Solitons and Fractals 5, 229 (1995).
O. Agam and N. Brenner, J. Phys. A 28, 1345 (1995).
O. Agam and S. Fishman, Phys. Rev. Lett. 73, 806 (1994); O. Agam and S. Fishman, J. Phys. A 26, 2113 (1993).
S. Fishman, B. Georgeot, and R. E. Prange, J. Phys. A 29, 919 (1996).
E. J. Heller, Wavepacket Dynamics and Quantum Chaology in Chaos and Quantum Physics, Eds. M. J. Giannoni, A. Voros, and J. Zinn-Justin, Elsevier Science Publishers, Amsterdam (1990).
A. I. Schnirelman, Usp. Mat. Nauk. 29, 181 (1974); Y. Colin de Verdiere, Commun. Math. Phys. 102, 497 (1985); S. Zelditch, Duke Math. J. 55, 919 (1987).
L. Kaplan and E. J. Heller, Linear and Nonlinear Theory of Eigenfunction Scars, Ann. Phys. (N. Y.), in press.
P. W. O’Connor and E. J. Heller, Phys. Rev. Lett. 61, 2288 (1988).
B. Li, Phys. Rev. E 55, 5376 (1997).
E. J. Heller, J. Chem. Phys. 72, 1337 (1980); E. J. Heller and M. J. Davis, J. Phys. Chem. 86, 2118 (1982); E. B. Stechel and E. J. Heller, Ann. Rev. Phys. Chem. 35, 563 (1984); E. J. Heller, Phys. Rev. A35, 1360 (1987).
L. Kaplan and E. J. Heller, Phys. Rev. Lett. 76, 1453 (1996); F.-M. Dittes, E. Doron, and U. Smilansky, Phys. Rev. E 49, R963 (1994).
N. L. Balazs and A. Voros, Europhys. Lett. 4, 1089 (1987); N. L. Balazs and A. Voros, Ann. Phys. (N. Y.) 190, 1 (1989); M. Saraceno, Ann. Phys. (N. Y.) 199, 37 (1990); M. Saraceno and A. Voros, Physica D 79, 206 (1994).
P. W. O’Connor, S. Tomsovic and E. J. Heller, Physica D 55, 340 (1992); P. W. O’Connor and S. Tomsovic, Ann. Phys. (N. Y.) 207, 218 (1991).
L. Kaplan and E. J. Heller, unpublished.
L. Kaplan, unpublished.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kaplan, L., Heller, E.J. (1999). Theory of Eigenfunction Scarring. 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_6
Download citation
DOI: https://doi.org/10.1007/978-1-4615-4875-1_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7212-7
Online ISBN: 978-1-4615-4875-1
eBook Packages: Springer Book Archive