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Linear and nonlinear theory of eigenfunction scars. (English) Zbl 0927.37016

The authors generalize in several ways the theory of scarring of eigenfunctions of classically chaotic systems by short periodic orbits. The contribution of nonlinear recurrences associated with homoclinic orbits to scarring is included and different scenarios of random and nonrandom long-time recurrences are considered. The importance of the local classical structure around the periodic orbit is emphasized. The crucial role of symmetry is also discussed. Quantitive measures of scarring are provided and comparisons are made with numerical data.
Reviewer: S.Musaev (Baku)

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C27 Periodic orbits of vector fields and flows
81Q50 Quantum chaos
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[1] Berry, M. V., Chaotic Behaviour of Deterministic Systems (1983), North-Holland: North-Holland Amsterdam, p. 171
[2] Bohigas, O.; Giannoni, M.-J.; Schmit, C., J. Phys. Lett., 45, L-1015 (1984)
[3] Heller, E. J., Phys. Rev. Lett., 53, 1515 (1984)
[4] O’Connor, P.; Gehlen, J. N.; Heller, E. J., Phys. Rev. Lett., 58, 1296 (1987)
[5] Sridhar, S., Phys. Rev. Lett., 67, 785 (1991)
[6] Stein, J.; Stöckman, H.-J., Phys. Rev. Lett., 68, 2867 (1992)
[7] Fromhold, T. M.; Wilkinson, P. B.; Sheard, F. W.; Eaves, L.; Miao, J.; Edwards, G., Phys. Rev. Lett., 75, 1142 (1995)
[8] Wintgen, D.; Honig, A., Phys. Rev. Lett., 63, 1467 (1989)
[9] Muller, K.; Wintgen, D., J. Phys. B, 27, 2693 (1994)
[10] Bogomolny, E. B., Physica D, 31, 169 (1988)
[11] Berry, M. V., (Giannoni, M.-J.; Voros, A.; Zinn-Justin, J., Les Houches Lecture Notes, Summer School on Chaos and Quantum Physics (1991), Elsevier Science: Elsevier Science Amsterdam)
[12] Aurich, R.; Steiner, F., Chaos, Solitons and Fractals, 5, 229 (1995) · Zbl 0817.58050
[13] Agam, O.; Brenner, N., J. Phys. A, 28, 1345 (1995) · Zbl 0853.47036
[14] Agam, O.; Fishman, S., Phys. Rev. Lett., 73, 806 (1994)
[15] Fishman, S.; Georgeot, B.; Prange, R. E., J. Phys. A, 29, 919 (1996)
[16] Nonnenmacher, S.; Voros, A., J. Phys. A, 30, 295 (1997) · Zbl 0922.58026
[17] Berry, M. V.; Tabor, M., Proc. Roy. Soc. A, 349, 101 (1976)
[18] O’Connor, P. W.; Heller, E. J., Phys. Rev. Lett., 61, 2288 (1988)
[19] Schnirelman, A. I., Usp. Mat. Nauk., 29, 181 (1974)
[20] Heller, E. J., Wavepacket Dynamics and Quantum Chaology, (Giannoni, M. J.; Voros, A.; Zinn-Justin, J., V-Chaos and Quantum Physics (1990), Elsevier Science: Elsevier Science Amsterdam)
[21] Li, B., Phys. Rev. E, 55, 5376 (1997)
[22] L. Kaplan, E. J. Heller, Weak quantum ergodicity, Physica D; L. Kaplan, E. J. Heller, Weak quantum ergodicity, Physica D · Zbl 0960.81024
[23] L. Kaplan, E. J. Heller, Phys. Rev. Lett. 76, 1996, 1453, Phys. Rev. E49; L. Kaplan, E. J. Heller, Phys. Rev. Lett. 76, 1996, 1453, Phys. Rev. E49
[24] L. Kaplan, E. J. Heller; L. Kaplan, E. J. Heller
[25] Balazs, N. L.; Voros, A., Europhys. Lett., 4, 1089 (1987)
[26] O’Connor, P. W.; Tomsovic, S.; Heller, E. J., Physica D, 55, 340 (1992)
[27] Heller, E. J., J. Chem. Phys., 72, 1337 (1980)
[28] L. Kaplan, Wave Function Intensity Statistics from Unstable Periodic Orbits, Phys. Rev. Lett.; L. Kaplan, Wave Function Intensity Statistics from Unstable Periodic Orbits, Phys. Rev. Lett.
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