Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times. (English) Zbl 1067.81040
Summary: We study the behaviour of time evolved quantum mechanical expectation values in Lagrangian states in the limit \(\hbar \rightarrow 0\) and \(t \rightarrow \infty\). We show that it depends strongly on the dynamical properties of the corresponding classical system. If the classical system is strongly chaotic, i.e. Anosov, then the expectation values tend to a universal limit. This can be viewed as an analogue of mixing in the classical system. If the classical system is integrable, then the expectation values need not converge, and if they converge their limit depends on the initial state. An additional difference occurs in the timescales for which we can prove this behaviour; in the chaotic case we get up to Ehrenfest time, \(t \sim \ln(1/\hbar)\), whereas for integrable system we have a much larger time range.
MSC:
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 37A60 | Dynamical aspects of statistical mechanics |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
References:
| [1] | Berry, M.V., Balazs, N.L.: Evolution of semiclassical quantum states in phase space. J. Phys. A 12(5), 625-642 (1979) · Zbl 0396.70018 · doi:10.1088/0305-4470/12/5/012 |
| [2] | Berry, M.V., Balazs, N.L., Tabor, M., Voros, A.: Quantum maps. Ann. Physics 122(1), 26-63 (1979) · doi:10.1016/0003-4916(79)90296-3 |
| [3] | Bonechi, F., De Bièvre, S.: Exponential mixing and | ln ?| time scales in quantized hyperbolic maps on the torus. Commun. Math. Phys. 211(3), 659-686 (2000) · Zbl 1053.81032 · doi:10.1007/s002200050831 |
| [4] | Bambusi, D., Graffi, S., Paul, T.: Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time. Asymptot. Anal. 21(2), 149-160 (1999) · Zbl 0934.35142 |
| [5] | Bouzouina, A., Robert, D.: Uniform semiclassical estimates for the propagation of quantum observables. Duke Math. J. 111(2), 223-252 (2002) · Zbl 1069.35061 |
| [6] | Brin, M., Stuck, G.: Introduction to dynamical systems. Cambridge: Cambridge University Press, 2002 · Zbl 1314.37002 |
| [7] | Bates, S., Weinstein, A.: Lectures on the geometry of quantization. Berkeley Mathematics Lecture Notes, Vol. 8, Providence, RI: Amer. Math. Soc. 1997 · Zbl 1049.53061 |
| [8] | Berman, G. P., Zaslavsky, G. M.: Condition of stochasticity in quantum nonlinear systems. Phys. A 91(3-4), 450-460 (1978) |
| [9] | Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497-502 (1985) · Zbl 0592.58050 |
| [10] | Chernov, N. I.: On Sinai-Bowen-Ruelle measures on horocycles or 3-D Anosov flows. Geom. Dedicata 68(3), 359-369 (1997) · Zbl 0898.58042 · doi:10.1023/A:1004948505533 |
| [11] | Chernov, N. I.: Markov approximations and decay of correlations for Anosov flows. Ann. of Math. (2) 147(2), 269-324 (1998) · Zbl 0911.58028 |
| [12] | Combescure, M., Robert, D.: Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. Asymptot. Anal. 14(4), 377-404 (1997) · Zbl 0894.35026 |
| [13] | De Bièvre, S., Robert, D.: Semiclassical propagation on | log ?| time scales. Int. Math. Res. Not. Vol. 12, 667-696 (2003) · Zbl 1127.81328 · doi:10.1155/S1073792803204268 |
| [14] | Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. of Math. (2) 147(2), 357-390 (1998) · Zbl 0911.58029 |
| [15] | Dimassi, M., Sjöstrand, J.: Spectral asymptotics in the semi-classical limit. London Mathematical Society Lecture Note Series, Vol. 268, Cambridge: Cambridge University Press, 1999 · Zbl 0926.35002 |
| [16] | Duistermaat, J. J.: Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 , 207-281 (1974) · Zbl 0285.35010 · doi:10.1002/cpa.3160270205 |
| [17] | Eberlein, P.: Geodesic flows in manifolds of nonpositive curvature. In: Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., Vol. 69, Providence, RI: Amer. Math. Soc. 2001, pp. 525-571 · Zbl 1030.53084 |
| [18] | Eskin, A., McMullen, C.: Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71(1), 181-209 (1993) · Zbl 0798.11025 · doi:10.1215/S0012-7094-93-07108-6 |
| [19] | Faure, F., Nonnenmacher, S., De Bièvre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239(3), 449-492 (2003) · Zbl 1033.81024 · doi:10.1007/s00220-003-0888-3 |
| [20] | Helffer, B., Martinez, A., Robert, D.: Ergodicité et limite semi-classique. Commun. Math. Phys. 109(2), 313-326 (1987) · Zbl 0624.58039 · doi:10.1007/BF01215225 |
| [21] | Hörmander, L.: The analysis of linear partial differential operators. I. second ed., Grundlehren der Mathematischen Wissenschaften, Vol. 256, Berlin: Springer-Verlag 1990 · Zbl 0712.35001 |
| [22] | Hörmander, L.: The analysis of linear partial differential operators. IV. Grundlehren der Mathematischen Wissenschaften, Vol. 275, Berlin: Springer-Verlag, 1985 · Zbl 0601.35001 |
| [23] | Ivrii, V.: Microlocal analysis and precise spectral asymptotics. Springer Monographs in Mathematics, Berlin: Springer-Verlag, 1998 · Zbl 0906.35003 |
| [24] | Liverani, C.: On contact Anosov flows. Ann. of Math. 159, 1275-1312 (2004) · Zbl 1067.37031 · doi:10.4007/annals.2004.159.1275 |
| [25] | O?Connor, P. W., Tomsovic, S., Heller, E. J.: Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier. Phys. D 55(3-4), 340-357 (1992) · Zbl 0753.58029 |
| [26] | Sinai, Ya. G.: Geodesic flows on manifolds of negative curvature. In: Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), New York: Plenum, 1995, pp. 201-215 · Zbl 0860.58032 |
| [27] | ?nirel?man, A. I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29, 6(180), 181-182 (1974) · Zbl 0324.58020 |
| [28] | Tomsovic, S., Heller, E. J.: Semiclassical dynamics of chaotic motion: unexpected long-time accuracy. Phys. Rev. Lett. 67(6), 664-667 (1991) · Zbl 0990.81511 · doi:10.1103/PhysRevLett.67.664 |
| [29] | Tomsovic, S., Heller, E. J.: Long-time semiclassical dynamics of chaos: the stadium billiard. Phys. Rev. E (3) 47(1), 282-299 (1993) |
| [30] | Zaslavsky, G. M.: Stochasticity in quantum systems. Phys. Rep. 80(3), 157-250 (1981) · doi:10.1016/0370-1573(81)90127-7 |
| [31] | Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919-941 (1987) · Zbl 0643.58029 · doi:10.1215/S0012-7094-87-05546-3 |
| [32] | Zelditch, S.: Quantum mixing. J. Funct. Anal. 140, 68-86 (1996) · Zbl 0858.58049 · doi:10.1006/jfan.1996.0098 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
