Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2525-2599.

We consider a nonlinear area preserving Anosov map M on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator M ^. The usual semi-classical Trace formula expresses TrM ^ t for finite time t, in the limit 0, in terms of periodic orbits of M of period t. Recent work reach time tt E /6 where t E =log(1/)/λ is the Ehrenfest time, and λ is the Lyapounov coefficient. Using a semi-classical normal form description of the dynamics uniformly over phase space, we show how to extend the trace formula for longer time of the form t=C.t E where C is any constant, with an arbitrary small error.

On considère une application M, Anosov non linéaire qui conserve l’aire sur le tore T 2 . C’est un des exemples les plus simples d’une dynamique chaotique. On s’intéresse à la dynamique quantique pour les temps longs, générée par un opérateur unitaire M ^. La formule des traces semi-classique habituelle exprime TrM ^ t pour t fini, dans la limite 0, en termes d’orbites périodiques de M de période t. Des travaux récents atteignent des temps tt E /6t E =log(1/)/λ est le temps d’Ehrenfest, et λ est le coefficient de Lyapounov. En utilisant une description uniforme de la dynamique au moyen d’une forme normale semi-classique, nous montrons comment étendre la formule des traces pour des temps plus longs, de la forme t=C.t E , où C est une constante arbitraire, et avec une erreur arbitrairement petite.

DOI: 10.5802/aif.2341
Classification: 81Q50, 37D20
Keywords: Quantum chaos, hyperbolic map, semiclassical trace formula, Ehrenfest time
Mot clés : chaos quantique, application hyperbolique, formule des traces semi-classique, temps d’Ehrenfest
Faure, Frédéric 1

1 Institut Fourier 100 rue des Maths, BP74 38402 St Martin d’Heres (France)
@article{AIF_2007__57_7_2525_0,
     author = {Faure, Fr\'ed\'eric},
     title = {Semi-classical formula beyond the {Ehrenfest} time in quantum chaos. {(I)} {Trace} formula},
     journal = {Annales de l'Institut Fourier},
     pages = {2525--2599},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     doi = {10.5802/aif.2341},
     zbl = {1145.81035},
     mrnumber = {2394550},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2341/}
}
TY  - JOUR
AU  - Faure, Frédéric
TI  - Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 2525
EP  - 2599
VL  - 57
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2341/
DO  - 10.5802/aif.2341
LA  - en
ID  - AIF_2007__57_7_2525_0
ER  - 
%0 Journal Article
%A Faure, Frédéric
%T Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula
%J Annales de l'Institut Fourier
%D 2007
%P 2525-2599
%V 57
%N 7
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2341/
%R 10.5802/aif.2341
%G en
%F AIF_2007__57_7_2525_0
Faure, Frédéric. Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2525-2599. doi : 10.5802/aif.2341. http://www.numdam.org/articles/10.5802/aif.2341/

[1] Anantharaman, N. Entropy and the localization of eigenfunctions (2004) (Ann. of Math.)

[2] Anantharaman, N.; Nonnenmacher, S. Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold (2006) (ArXiv Mathematical Physics e-prints, math-ph/0610019)

[3] Arnold, V. I. Geometrical methods in the theory of ordinary differential equations, Springer Verlag, 1988 | MR | Zbl

[4] Bambusi, D.; Graffi, S.; Paul, T. Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time, Asymptot. Anal., Volume 21 (1999) no. 2, pp. 149-160 | MR | Zbl

[5] Bohigas, O. Random matrix theories and chaotic dynamics, Chaos and Quantum Physics, Proceedings of the Les Houches Summer School (1989), Volume 45 (1991), pp. 87-199 | MR

[6] Bohigas, O.; Giannoni, M. J.; Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett., Volume 52 (1984) no. 1, pp. 1-4 | DOI | MR | Zbl

[7] Bonechi, F.; DeBièvre, S. Exponential mixing and ln(h) timescales in quantized hyperbolic maps on the torus, Comm. Math. Phys., Volume 211 (2000), pp. 659-686 | DOI | MR | Zbl

[8] Bouclet, J. M.; DeBièvre, S. Long time propagation and control on scarring for perturbated quantized hyperbolic toral automorphisms, Annales Henri Poincaré, Volume 6 (2005) no. 5, pp. 885-913 | DOI | MR | Zbl

[9] Bouzouina, A.; Robert, D. Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., Volume 111 (2002) no. 2, pp. 223-252 | DOI | MR | Zbl

[10] Cargo, M.; Gracia-Saz, A.; Littlejohn, R. G.; Reinsch, M. W.; Rios, P. M. Quantum normal forms, moyal star product and bohr-sommerfeld approximation, J. Phys. A: Math. Gen., Volume 38 (2005), pp. 1997-2004 | DOI | MR | Zbl

[11] Colin de Verdière, Y. Ergodicité et fonctions propres du laplacien. (Ergodicity and eigenfunctions of the Laplacian), Commun. Math. Phys., Volume 102 (1985), pp. 497-502 | MR | Zbl

[12] Colin de Verdière, Y.; Parisse, B. Équilibre instable en régime semi-classique - I. Concentration microlocale, Communications in Partial Differential Equations, Volume 19 (1994) no. 9–10, pp. 1535-1563 | DOI | MR | Zbl

[13] Colin de Verdière, Y.; Parisse, B. Équilibre instable en régime semi-classique - II. Conditions de Bohr-Sommerfeld, Annales de l’Institut Henri Poincaré- Physique Théorique, Volume 61 (1994) no. 3, pp. 347-367 | Numdam | Zbl

[14] Colin de Verdière, Y.; Parisse, B. Singular bohr-sommerfeld rules, Commun. Math. Phys, Volume 205 (1999), pp. 459-500 | DOI | MR | Zbl

[15] Combescure, M.; Ralston, J.; Robert, D. A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Commun. Math. Phys., Volume 202 (1999) no. 2, pp. 463-480 | DOI | MR | Zbl

[16] Combescure, M.; Robert, D. Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptotic Anal., Volume 14 (1997) no. 4, pp. 377-404 | MR | Zbl

[17] De Bièvre, S. Recent results on quantum map eigenstates, Mathematical physics of quantum mechanics (Lecture Notes in Phys.), Volume 690, Springer, Berlin, 2006, pp. 367-381 | MR | Zbl

[18] De Bièvre, Stephan Quantum chaos: a brief first visit, Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000) (Contemp. Math.), Volume 289, Amer. Math. Soc., 2001, pp. 161-218 | MR | Zbl

[19] DeLatte, D. Nonstationnary normal forms and cocycle invariants, Random and Computational dynamics, Volume 1 (1992), pp. 229-259 | MR | Zbl

[20] DeLatte, D. On normal forms in hamiltonian dynamics, a new approach to some convergence questions, Ergod. Th. and Dynam. Sys., Volume 15 (1995), pp. 49-66 | DOI | MR | Zbl

[21] Dimassi, M.; Sjöstrand, J. Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Notes, 268, Cambridge University Press, 1999 | MR | Zbl

[22] Eckhardt, B.; Fishman, S.; Keating, J.; Agam, O.; Main, J.; Müller, K. Approach to ergodicity in quantum wave functions, Phys. Rev. E, Volume 52 (1995), pp. 5893-5903 | DOI

[23] Evans, L.; Zworski, M. Lectures on semiclassical analysis, 2003 (http://math.berkeley.edu/ zworski/)

[24] Faure, F. Semiclassical formula beyond the ehrenfest time in quantum chaos. (II) propagator formula, 2006 (in preparation)

[25] Faure, F. Prequantum chaos: Resonances of the prequantum cat map, Journal of Modern Dynamics, Volume 1 (2007) no. 2, pp. 255-285 | DOI | MR

[26] Faure, F.; Nonnenmacher, S. On the maximal scarring for quantum cat map eigenstates, Communications in Mathematical Physics, Volume 245 (2004), pp. 201-214 | DOI | MR | Zbl

[27] Faure, F.; Nonnenmacher, S.; DeBièvre, S. Scarred eigenstates for quantum cat maps of minimal periods, Communications in Mathematical Physics, Volume 239 (2003), pp. 449-492 | DOI | MR | Zbl

[28] Folland, G. B. Harmonic Analysis in phase space, Princeton University Press, 1989 | MR | Zbl

[29] Giannoni, M. J.; Voros, A.; Zinn-Justin, J. Chaos and Quantum Physics, Les Houches Session LII 1989, North-Holland, 1991 | MR

[30] Gohberg, I.; Goldberg, S.; Krupnik, N. Traces and Determinants of Linear Operators, Birkhauser, 2000 | MR | Zbl

[31] Gracia-Saz, A. The symbol of a function of a pseudo-differential operator., Annales de l’Institut Fourier, Volume 55 (2005) no. 7, pp. 2257-2284 | DOI | Numdam | Zbl

[32] Guillemin, V. Wave-trace invariants, Duke Math. J., Volume 83 (1996) no. 2, pp. 287-352 | DOI | MR | Zbl

[33] Gutzwiller, M. Periodic orbits and classical quantization conditions, J. Math. Phys., Volume 12 (1971), pp. 343-358 | DOI

[34] Gutzwiller, M. Chaos in classical and quantum mechanics, Springer-Verlag, 1991 | MR | Zbl

[35] Haake, F. Quantum Signatures of Chaos, Springer, 2001 | MR | Zbl

[36] Hagedorn, G. A.; Joye, A. Exponentially acurrate semiclassical dynamics: Propagation, localization, ehrenfest times, scattering, and more general states, Ann. Henri Poincaré, Volume 1 (2000), pp. 837-883 | DOI | MR | Zbl

[37] Hasselblatt, B. Hyperbolic dynamics, Handbook of Dynamical Systems, North Holland, Volume 1A (2002), pp. 239-320 | DOI | MR | Zbl

[38] Heller, E. J. Time dependant approach to semiclassical dynamics, J. Chem. Phys., Volume 62 (1975), pp. 1544-1555 | DOI

[39] Hepp, K. The classical limit of quantum mechanical correlation funtions, Comm. Math. Phys., Volume 35 (1974), pp. 265-277 | DOI | MR

[40] Hurder, S.; Katok, A. Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math., Inst. Hautes Étud. Sci., Volume 72 (1990), pp. 5-61 | DOI | Numdam | MR | Zbl

[41] Iantchenko, A. The Birkhoff normal form for a Fourier integral operator. (La forme normale de Birkhoff pour un opérateur intégral de Fourier.), Asymptotic Anal., Volume 17 (1998) no. 1, pp. 71-92 | MR | Zbl

[42] Iantchenko, A.; Sjöstrand, J. Birkhoff normal forms for Fourier integral operators. II, Am. J. Math., Volume 124 (2002) no. 4, pp. 817-850 | DOI | MR | Zbl

[43] Iantchenko, A.; Sjöstrand, J.; Zworski, M. Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett., Volume 9 (2002) no. 2-3, pp. 337-362 | MR | Zbl

[44] Joye, A.; Hagedorn, G. Semiclassical dynamics with exponentially small error estimates, Comm. in Math. Phys., Volume 207 (1999), pp. 439-465 | DOI | MR | Zbl

[45] Katok, A.; Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995 | MR | Zbl

[46] Keating, J. P. Asymptotic properties of the periodic orbits of the cat maps, Nonlinearity, Volume 4 (1991), pp. 277-307 | DOI | MR | Zbl

[47] Keating, J. P. The cat maps: quantum mechanics and classical motion, Nonlinearity, Volume 4 (1991), pp. 309-341 | DOI | MR | Zbl

[48] Littlejohn, R. G. The semiclassical evolution of wave-packets, Phys. Rep., Volume 138 (1986) no. 4–5, pp. 193-291 | DOI | MR

[49] Martinez, A. An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer, New York, 2002 | MR | Zbl

[50] Nonnenmacher, S. Evolution of lagrangian states through pertubated cat maps, Preprint, 2004

[51] Perelomov, A. Generalized coherent states and their applications, Springer-Verlag, 1986 | MR | Zbl

[52] Pollicott, M.; Yuri, M. Dynamical Systems and Ergodic theory, Cambridge University Press, 1998 | MR | Zbl

[53] Schubert, R. Semi-classical behaviour of expectation values in time evolved lagrangian states for large times, Commun. Math. Phys., Volume 256 (2005), pp. 239-254 | DOI | MR | Zbl

[54] Sjöstrand, J. Resonances associated to a closed hyperbolic trajectory in dimension 2, Asymptotic Anal., Volume 36 (2003) no. 2, pp. 93-113 | MR | Zbl

[55] Sjöstrand, J.; Zworski, M. Quantum monodromy and semi-classical trace formulae, J. Math. Pures Appl., Volume 1 (2002), pp. 1-33 | MR | Zbl

[56] Tomsovic, S.; Heller, E. J. Long-time semi-classical dynamics of chaos: the stadium billard, Physical Review E, Volume 47 (1993), pp. 282 | DOI | MR

[57] Zelditch, S. Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Rev. Mod. Phys., Volume 55 (1987), pp. 919-941 | MR | Zbl

[58] Zelditch, S. Quantum dynamics from the semi-classical viewpoint, Lectures at I.H.P., 1996 (http://mathnt.mat.jhu.edu/zelditch)

[59] Zelditch, S. Wave invariants at elliptic closed geodesics, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 145-213 | DOI | MR | Zbl

[60] Zelditch, S. Wave invariants for non-degenerate closed geodesics, Geom. Funct. Anal., Volume 8 (1998) no. 1, pp. 179-217 | DOI | MR | Zbl

[61] Zelditch, S. Quantum ergodicity and mixing of eigenfunctions, Elsevier Encyclopedia of Math. Phys, 2005

[62] Zhang, W. M.; Feng, D. H.; Gilmore, R. Coherent states: theory and some applications, Rev. Mod. Phys., Volume 62 (1990), pp. 867 | DOI | MR

Cited by Sources: