Resummation and the semiclassical theory of spectral statistics

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 463, Issue 2088, 2007, Pages 3241-3250

  Keating, Jonathan P ^a   Müller, Sebastian ^b  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Periodic orbit theory; Quantum chaos; Random matrix theory; Semiclassical approximation; Spectral statistics

Indexed keywords

EID: 36349015525     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2007.0178     Document Type: Article

References (29)

1. Andreev, A. V., Agam, O., Simons, B. D. & Altshuler, B. L. 1996 Quantum chaos, irreversible classical dynamics, and random matrix theory. Phys. Rev. Lett. 76, 3947-3950. (doi:10.1103/PhysRevLett.76.3947)
2. Argaman, N., Dittes, F.-M., Doron, E., Keating, J. P., Kitaev, A. Y., Sieber, M. & Smilansky, U. 1993 Correlations in the actions of periodic orbits derived from quantum chaos. Phys. Rev. Lett. 71, 4326-4329. (doi:10.1103/PhysRevLett.71.4326)
3. Berry, M. V. 1985 Semiclassical theory of spectral rigidity. Proc. R. Soc. A 400, 229-251. (doi:10.1098/rspa.1985.0078)
4. Berry, M. V. 1986 Riemann's zeta function: a model for quantum chaos? In Quantum chaos and statistical nuclear physics (eds T. H. Seligman & H. Nishioka). Springer Lecture Notes in Physics, no. 263, pp. 1-17. Berlin, Germany: Springer.
5. Berry, M. V. 1987 The Bakerian lecture, 1987: quantum chaology. Proc. R. Soc. A 413, 183-198. (doi:10.1098/rspa.1987.0109)
6. Berry, M. V. & Keating, J. P. 1990 A rule for quantizing chaos? J. Phys. A 23, 4839-4849. (doi:10.1088/0305-4470/23/21/024)
7. Berry, M. V. & Keating, J. P. 1992 A new asymptotic representation for ζ((1/2)+it) and quantum spectral determinants. Proc. R. Soc. A 437, 151-173. (doi:10.1098/rspa.1992.0053)
8. Bogomolny, E. B. & Keating, J. P. 1996 Gutzwiller's trace formula and spectral statistics: beyond the diagonal approximation. Phys. Rev. Lett. 77, 1472-1475. (doi:10.1103/PhysRevLett.77.1472)
9. Bogomolny, E. B., Georgeot, B., Giannoni, M. J. & Schmit, C. 1992 Chaotic billiards generated by arithmetic groups. Phys. Rev. Lett. 69, 1477-1480. (doi:10.1103/PhysRevLett.69.1477)
10. Bohigas, O., Giannoni, M. J. & Schmit, C. 1984 Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1-4. (doi:10.1103/PhysRevLett.52.1)
11. Casati, G., Valz-Gris, F. & Guarneri, I. 1980 On the connection between quantization of non-integrable systems and statistical theory of spectra. Lett. Nuovo Cimento 28, 279.
12. Gutzwiller, M. C. 1990 Chaos in classical and quantum mechanics. New York, NY: Springer.
13. Haake, F. 2001 Quantum signatures of chaos, 2nd edn. Berlin, Germany: Springer.
14. Hannay, J. H. & Ozorio de Almeida, A. M. 1984 Periodic orbits and a correlation function for the semiclassical density of states. J. Phys. A 17, 3429-3440. (doi:10.1088/0305-4470/17/18/013)
15. Heusler, S., Müller, S., Altland, A., Braun, P. & Haake, F. 2007 Periodic-orbit theory of level correlations. Phys. Rev. Lett. 98, 044103. (doi:10.1103/PhysRevLett.98.044103)
16. Keating, J. P. 1991 The cat maps: quantum mechanics and classical motion. Nonlinearity 4, 309-341. (doi:10.1088/0951-7715/4/2/006)
17. Keating, J. P. 1992 Periodic orbit resummation and the quantization of chaos. Proc. R. Soc. A 436, 99-108. (doi:10.1098/rspa.1992.0007)
18. Keating, J. P. & Mezzadri, F. 2000 Pseudo-symmetries of Anosov maps and spectral statistics. Nonlinearity 13, 747-775. (doi:10.1088/0951-7715/13/3/313)
19. Keating, J. P. & Sieber, M. 1994 Calculation of spectral determinants. Proc. R. Soc. A 447, 413-437. (doi:10.1098/rspa. 1994.0148)
20. Ketteman, S., Klakow, D. & Smilansky, U. 1997 Characterization of quantum chaos by the autocorrelation function of spectral determinants. J. Phys. A 30, 3643-3662. (doi:10.1088/0305-4470/30/10/035)
21. McDonald, S. W. & Kaufman, A. N. 1979 Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories. Phys. Rev. Lett. 42, 1189-1191. (doi:10.1103/PhysRevLett.42.1189)
22. Müller, S., Heusler, S., Braun, P., Haake, F. & Altland, A. 2004 Semiclassical foundation of universality in quantum chaos. Phys. Rev. Lett. 93, 014103. (doi:10.1103/PhysRevLett.93.014103)
23. Müller, S., Heusler, S., Braun, P., Haake, F. & Altland, A. 2005 Periodic-orbit theory of universality in quantum chaos. Phys. Rev. E 72, 046207. (doi:10.1103/PhysRevE.72.046207)
24. Müller, J., Micklitz, T. & Altland, A. 2007 Universal spectral correlations from the ballistic sigma model. (http://arXiv.org/abs/0707.2274)
25. Muzykantskii, B. A. & Khmel'nitskii, D. E. 1995 Effective action in the theory of quasi-ballistic disordered conductors. JETP Lett. 62, 76-82.
26. Sieber, M. 2002 Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems. J. Phys. A 35, L613-L619. (doi:10.1088/0305-4470/35/42/104)
27. Sieber, M. & Richter, K. 2001 Correlations between periodic orbits and their role in spectral statistics. Phys. Scripta T 90, 128-133. (doi:10.1238/Physica.Topical.090a00128)
28. Stöckmann, H.-J. 1999 Quantum chaos: an introduction. Cambridge, UK: Cambridge University Press.
29. Wigner, E. P. 1959 Statistical properties of real symmetric matrices with many dimensions. In Proc. 4th Canadian Mathematical Congress, Toronto, pp. 174-184. Toronto, Canada: University of Toronto Press.