Long-time dynamics of the perturbed Schrödinger equation on negatively curved surfaces. (English) Zbl 1344.81099
Summary: We consider perturbations of the semiclassical Schrödinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 53C20 | Global Riemannian geometry, including pinching |
References:
| [1] | Abraham R.: Lectures of Smale on Differential Topology. Lectures at Columbia University, New York (1962) |
| [2] | Anantharaman N.: Entropy and the localization of eigenfunctions. Ann. Math. 168, 435-475 (2008) · Zbl 1175.35036 · doi:10.4007/annals.2008.168.435 |
| [3] | Anantharaman N., Nonnenmacher S.: Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Festival Yves Colin de Verdière. Ann. Inst. Fourier (Grenoble) 57, 2465-2523 (2007) · Zbl 1145.81033 · doi:10.5802/aif.2340 |
| [4] | Anantharaman N., Rivière G.: Dispersion and controllability for the Schrödinger equation on negatively curved manifolds. Anal. PDE 5, 313-338 (2012) · Zbl 1267.35176 · doi:10.2140/apde.2012.5.313 |
| [5] | Anosov, D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90 (1967) · Zbl 0176.19101 |
| [6] | Bambusi D., Graffi S., Paul T.: Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time. Asymptot. Anal. 21, 149-160 (1999) · Zbl 0934.35142 |
| [7] | Barreira L., Wolf C.: Dimension and ergodic decompositions for hyperbolic flows. Discret. Contin. Dyn. Syst. 17, 201-212 (2007) · Zbl 1144.37012 |
| [8] | Bolte J., Schwaibold T.: Stability of wave packet dynamics under perturbations. Phys. Rev. E 73, 026223 (2006) · doi:10.1103/PhysRevE.73.026223 |
| [9] | Bonechi F., De Bièvre S.: Exponential mixing and \[{|{\rm log} \hbar|}\]|logħ| time scales in quantized hyperbolic maps on the torus. Commun. Math. Phys. 211, 659-686 (2000) · Zbl 1053.81032 · doi:10.1007/s002200050831 |
| [10] | Bouclet J.M., De Bièvre S.: Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms. Ann. Henri Poincaré 6, 885-913 (2005) · Zbl 1088.81049 · doi:10.1007/s00023-005-0228-6 |
| [11] | Bouzouina A., Robert D.: Uniform semiclassical estimates for the propagation of quantum observables. Duke Math. J. 111, 223-252 (2002) · Zbl 1069.35061 · doi:10.1215/S0012-7094-02-11122-3 |
| [12] | Brooks S., Lindenstrauss E.: Joint quasimodes, positive entropy, and quantum unique ergodicity. Invent. Math. 198, 219-259 (2014) · Zbl 1343.58016 · doi:10.1007/s00222-014-0502-7 |
| [13] | Burq, N.: Mesures semi-classiques et mesures de défaut. Sem. Bourbaki 1996-1997, Exp. 826, 167-195 (1997) · Zbl 0954.35102 |
| [14] | Canzani Y., Jakobson D., Toth J.: On the distribution of perturbations of propagated Schrödinger eigenfunctions. J. Spectr. Theory 4, 283-307 (2014) · Zbl 1296.35102 · doi:10.4171/JST/70 |
| [15] | Cartan H.: Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications. Ann. Sci. ENS 45, 255-346 (1928) · JFM 54.0357.06 |
| [16] | Colin de Verdière Y.: Ergodicité et fonctions propres du Laplacien. Commun. Math. Phys. 102, 497-502 (1985) · Zbl 0576.58032 · doi:10.1007/BF01209296 |
| [17] | Combescure M., Robert D.: Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. Asymptot. Anal. 14, 377-404 (1997) · Zbl 0894.35026 |
| [18] | Combescure M., Robert D.: A phase-space study of the quantum Loschmidt Echo in the semiclassical limit. Ann. Henri Poincaré 8, 91-108 (2007) · Zbl 1109.81051 · doi:10.1007/s00023-006-0301-9 |
| [19] | de la Llave R., Marco J.M., Moriyon R.: Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. Ann. Math. 123, 537-611 (1986) · Zbl 0603.58016 · doi:10.2307/1971334 |
| [20] | Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series, vol. 268 · Zbl 0926.35002 |
| [21] | Dyatlov S., Guillarmou C.: Microlocal limits of plane waves and Eisenstein functions. Ann. Sci. Éc. Norm. Supér. 47, 371-448 (2014) · Zbl 1297.58007 |
| [22] | Eels Jr., J.: On the geometry of function spaces. Symp. Inter. de Topología Alg. Mexico, Universidad Nacional Autónoma de México y la Unesco, pp. 303-308 (1958) · Zbl 0603.58016 |
| [23] | Eels J. Jr.: A setting for global analysis. Bull. Am. Math. Soc. 72(5), 751-807 (1966) · doi:10.1090/S0002-9904-1966-11558-6 |
| [24] | Eswarathasan, S., Rivière, G.: Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces. J. Inst. Math. Jussieu. Published online 27 August 2015. doi:10.1017/S1474748015000262 · Zbl 1379.37129 |
| [25] | Eswarathasan S., Toth J.: Average pointwise bounds for deformations of Schrodinger eigenfunctions. Ann. Henri Poincaré 14, 611-637 (2012) · Zbl 1263.81186 · doi:10.1007/s00023-012-0198-4 |
| [26] | Furstenberg, H.: The unique ergodicity of the horocycle flow. Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund). In: Lecture Notes in Mathematics, vol. 318, pp. 95-115. Springer, Berlin (1973) · Zbl 1069.35061 |
| [27] | Gérard, P.: Mesures semi-classiques et ondes de Bloch. Sem. EDP (Polytechnique) 1990-1991, Exp. 16 (1991) · Zbl 1234.58007 |
| [28] | Gorin T., Prosen T., Seligman T.H., Zdinaric M.: Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 435, 33-156 (2006) · doi:10.1016/j.physrep.2006.09.003 |
| [29] | Goussev, A., Jalabert, R.A., Pastawski, H.M., Wisniacki, D.: Loschmidt Echo. Scholarpedia 7(8), 11687 (2012). arXiv:1206.6348 · Zbl 0322.28012 |
| [30] | Helffer B., Martinez A., Robert D.: Ergodicité et limite semi-classique. Commun. Math. Phys. 109, 313-326 (1987) · Zbl 0624.58039 · doi:10.1007/BF01215225 |
| [31] | Hirsch M., Pugh C.: Smoothness of horocycle foliations. J. Differ. Geom. 10, 225-238 (1975) · Zbl 0312.58008 |
| [32] | Hörmander L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1985) · Zbl 0601.35001 |
| [33] | Jacquod P., Petitjean C.: Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom. Adv. Phys. 58, 67-196 (2009) · doi:10.1080/00018730902831009 |
| [34] | Jacquod P., Silvestrov P., Beenakker C.: Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo. Phys. Rev. E 64, 055203(R) (2001) · doi:10.1103/PhysRevE.64.055203 |
| [35] | Jalabert R.A., Pastawski H.M.: Environment-independent decoherence rate in classically chaotic systems. Phys. Rev. Lett. 86, 2490 (2001) · doi:10.1103/PhysRevLett.86.2490 |
| [36] | Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995) · Zbl 0878.58020 |
| [37] | Lang S.: Introduction to Differentiable Manifolds, 2nd edn. Springer, New York (2002) · Zbl 1008.57001 |
| [38] | Macià, F., Rivière, G.: Concentration and non concentration for the Schrödinger evolution on Zoll manifolds. Commun. Math. Phys. (2015) (in press). doi:10.1007/s00220-015-2504-8 · Zbl 1348.58017 |
| [39] | Marcus B.: Ergodic properties of horocycle flows for surfaces of negative curvature. Ann. Math. 105, 81-105 (1977) · Zbl 0322.28012 · doi:10.2307/1971026 |
| [40] | Moser J.: On a theorem of Anosov. Differ. Equ. 5, 411-440 (1969) · Zbl 0169.42303 · doi:10.1016/0022-0396(69)90083-7 |
| [41] | Nonnenmacher S.: Anatomy of quantum chaotic eigenstates. Chaos Prog. Math. Phys. 66, 193-238 (2013) · Zbl 1342.81155 · doi:10.1007/978-3-0348-0697-8_6 |
| [42] | Peres A.: Stability of quantum motion in chaotic and regular systems. Phys. Rev. A 30, 1610-1615 (1984) · doi:10.1103/PhysRevA.30.1610 |
| [43] | Ruggiero, R.O.: Dynamics and global geometry of manifolds without conjugate points. Ensaios Mat. 12. Soc. Bras. Mat. (2007) · Zbl 1133.37009 |
| [44] | Sarnak P.: Recent progress on the quantum unique ergodicity conjecture. Bull. AMS 48, 211-228 (2011) · Zbl 1234.58007 · doi:10.1090/S0273-0979-2011-01323-4 |
| [45] | Schubert R.: Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times. Commun. Math. Phys. 256, 239-254 (2005) · Zbl 1067.81040 · doi:10.1007/s00220-005-1319-4 |
| [46] | Shnirelman A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk. 29, 181-182 (1974) · Zbl 0324.58020 |
| [47] | Smale S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861-866 (1965) · Zbl 0143.35301 · doi:10.2307/2373250 |
| [48] | Smale S.: Differentiable dynamical systems. Bull. AMS 73, 747-817 (1967) · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1 |
| [49] | Zelditch S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919-941 (1987) · Zbl 0643.58029 · doi:10.1215/S0012-7094-87-05546-3 |
| [50] | Zelditch, S.: Recent developments in mathematical quantum chaos. In: Current Developments in Mathematics 2009, pp. 115-204. International Press, Somerville (2010) · Zbl 1223.37113 |
| [51] | Zworski, M.: Semiclassical analysis. In: Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence (2012) · Zbl 1252.58001 |
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.
