Advertisement

Communications in Mathematical Physics

, Volume 304, Issue 1, pp 1–48 | Cite as

From Open Quantum Systems to Open Quantum Maps

  • Stéphane Nonnenmacher
  • Johannes Sjöstrand
  • Maciej ZworskiEmail author
Open Access
Article

Abstract

For a class of quantized open chaotic systems satisfying a natural dynamical assumption we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories.

Keywords

Neighbourhood Versus Pseudodifferential Operator Open Quantum Principal Symbol Open Quantum System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We would like to thank the National Science Foundation for partial support under the grant DMS-0654436. This article was completed while the first author was visiting the Institute of Advanced Study in Princeton, supported by the National Science Foundation under agreement No. DMS-0635607. The first and second authors were also partially supported by the Agence Nationale de la Recherche under the grant ANR -09-JCJC-0099-01. Thanks also to Edward Ott for his permission to include Fig. 3 in our paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Aguilar J., Combes J.M.: A class of analytic perturbations for one-body Schröfdinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971)CrossRefzbMATHADSMathSciNetGoogle Scholar
  2. 2.
    Alexandrova I.: Semi-Classical Wavefront Set and Fourier Integral Operators. Can. J. Math 60, 241–263 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Balazs N.L., Voros A.: The quantized baker’s transformation. Ann. Phys. (NY) 190, 1–31 (1989)CrossRefzbMATHADSMathSciNetGoogle Scholar
  4. 4.
    Bogomolny E.B.: Semiclassical quantization of multidimensional systems. Nonlinearity 5, 805–866 (1992)CrossRefzbMATHADSMathSciNetGoogle Scholar
  5. 5.
    Borgonovi F., Guarneri I., Shepelyansky D.L.: Statistics of quantum lifetimes in a classically chaotic system. Phys. Rev. A 43, 4517–4520 (1991)CrossRefADSGoogle Scholar
  6. 6.
    Bony J.-M., Chemin J.-Y.: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. math. France 122, 77–118 (1994)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Borgonovi F., Guarneri I., Shepelyansky D.L.: Statistics of quantum lifetimes in a classically chaotic system. Phys. Rev. A 43, 4517–4520 (1991)CrossRefADSGoogle Scholar
  8. 8.
    Bowen R., Walters P.: Expansive One-parameter Flows. J. Diff. Eq. 12, 180–193 (1972)CrossRefzbMATHADSMathSciNetGoogle Scholar
  9. 9.
    Chirikov, B.V., Izrailev, F.M., Shepelyansky, D.L.: Dynamical stochasticity in classical and quantum mechanics. Math. Phys. Rev., 2, 209–267 (1981); Soviet Sci. Rev. Sect. 2 C: Math. Phys. Rev. 2, Chur: Harwood Academic, 1981Google Scholar
  10. 10.
    Cvitanović P., Rosenquist P., Vattay G., Rugh H.H.: A Fredholm determinant for semiclassical quantization. CHAOS 3, 619–636 (1993)CrossRefzbMATHADSMathSciNetGoogle Scholar
  11. 11.
    Degli Esposti, M., Graffi, S. (eds): The mathematical aspects of quantum maps. Heidelberg: Springer, 2003Google Scholar
  12. 12.
    Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the semi-classical limit. Cambridge: Cambridge University Press, 1999Google Scholar
  13. 13.
    Doron, E., Smilansky, U.: Semiclassical quantization of chaotic billiards: a scattering theory approach. Nonlinearity 5, 1055–1084 (1992); Rouvinez, C., Smilansky, U.: A scattering approach to the quantization of Hamiltonians in two dimensions – application to the wedge billiard. J. Phys. A 28, 77–104 (1995)Google Scholar
  14. 14.
    Burq N., Guillarmou C., Hassell A.: Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics. GAFA 20, 627–656 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Evans, L.C., Zworski, M.: Lectures on Semiclassical Analysis http://math.berkeley.edu/~zworski/semiclassical.pdf
  16. 16.
    Georgeot B., Prange R.E.: Fredholm theory for quasiclassical scattering. Phys. Rev. Lett. 74, 4110–4113 (1995)CrossRefzbMATHADSMathSciNetGoogle Scholar
  17. 17.
    Gaspard P., Rice S.A.: Semiclassical quantization of the scattering from a classically chaotic repellor. J. Chem. Phys. 90, 2242–2254 (1989)CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Gérard C.: Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes. Mémoires de la Société Mathématique de France Sér. 2(31), 1–146 (1988)Google Scholar
  19. 19.
    Gérard C., Sjöstrand J.: Semiclassical resonances generated by a closed trajectory of hyperbolic type. Commun. Math. Phys. 108, 391–421 (1987)CrossRefzbMATHADSGoogle Scholar
  20. 20.
    Gutzwiller M.: Chaos in classical and quantum mechanics. Springer, New York (1990)zbMATHGoogle Scholar
  21. 21.
    Helffer B., Sjöstrand J.: Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.) 24, 1–228 (1986)Google Scholar
  22. 22.
    Hörmander L.: The Analysis of Linear Partial Differential Operators. Vol. I, II. Springer-Verlag, Berlin (1983)CrossRefGoogle Scholar
  23. 23.
    Hörmander L.: The Analysis of Linear Partial Differential Operators. Vol. III, IV. Springer-Verlag, Berlin (1985)Google Scholar
  24. 24.
    Katok A., Hasselblatt B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  25. 25.
    Keating J.P., Novaes M., Prado S.D., Sieber M.: Semiclassical structure of quantum fractal eigenstates. Phys. Rev. Lett. 97, 150406 (2006)CrossRefADSGoogle Scholar
  26. 26.
    Martinez, A.: Resonance free domains for non globally analytic potentials. Ann. Henri Poincaré 3(4), 739–756 (2002). Erratum: Ann. Henri Poincaré 8(7), 1425–1431 (2007)Google Scholar
  27. 27.
    Nonnenmacher S., Rubin M.: Resonant eigenstates for a quantized chaotic system. Nonlinearity 20, 1387–1420 (2007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  28. 28.
    Nonnenmacher S., Zworski M.: Distribution of resonances for open quantum maps. Commun. Math. Phys. 269, 311–365 (2007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  29. 29.
    Nonnenmacher S., Zworski M.: Fractal Weyl laws in discrete models of chaotic scattering. J. Phys. A 38, 10683–10702 (2005)CrossRefzbMATHADSMathSciNetGoogle Scholar
  30. 30.
    Nonnenmacher S., Zworski M.: Quantum decay rates in chaotic scattering. Acta Math. 203, 149–233 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Nonnenmacher, S., Sjöstrand, J., Zworski, M.: Fractal Weyl law for open quantum chaotic maps. In preparationGoogle Scholar
  32. 32.
    Ozorio de Almeida A.M., Vallejos R.O.: Decomposition of Resonant Scatterers by Surfaces of Section. Ann. Phys. (NY) 278, 86–108 (1999)CrossRefzbMATHADSMathSciNetGoogle Scholar
  33. 33.
    Petkov V., Zworski M.: Semi-classical estimates on the scattering determinant. Ann. H. Poincaré 2, 675–711 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Poon L., Campos J., Ott E., Grebogi C.: Wada basin boundaries in chaotic scattering. Int. J. Bifurcation and Chaos 6, 251–266 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Prosen T.: General quantum surface-of-section method. J. Phys. A28, 4133–4155 (1995)ADSMathSciNetGoogle Scholar
  36. 36.
    Saraceno M., Vallejos R.O.: The quantized D-transformation. Chaos 6, 193–199 (1996)CrossRefzbMATHADSMathSciNetGoogle Scholar
  37. 37.
    Schomerus H., Tworzydlo J.: Quantum-to-classical crossover of quasi-bound states in open quantum systems. Phys. Rev. Lett. 93, 154102 (2004)CrossRefADSGoogle Scholar
  38. 38.
    Sjöstrand J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60, 1–57 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Sjöstrand, J.: A trace formula and review of some estimates for resonances. In: Microlocal analysis and spectral theory (Lucca, 1996), 377–437, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 490, Dordrecht: Kluwer Acad. Publ., 1997, pp. 377–437Google Scholar
  40. 40.
    Sjöstrand, J.: Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations. http://arxiv.org/abs/0802.3584v3 [math.sp], 2009
  41. 41.
    Sjöstrand J., Zworski M.: Complex scaling and the distribution of scattering poles. J. AMS 4, 729–769 (1991)zbMATHGoogle Scholar
  42. 42.
    Sjöstrand J., Zworski M.: Quantum monodromy and semiclassical trace formulae. J. Math. Pure Appl. 81, 1–33 (2002)CrossRefzbMATHGoogle Scholar
  43. 43.
    Sjöstrand J., Zworski M.: Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137, 381–459 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Sjöstrand J., Zworski M.: Elementary linear algebra for advanced spectral problems. Ann. l’Inst. Fourier 57, 2095–2141 (2007)zbMATHCrossRefGoogle Scholar
  45. 45.
    Tang S.H., Zworski M.: From quasimodes to resonances. Math. Res. Lett. 5, 261–272 (1998)MathSciNetGoogle Scholar
  46. 46.
    Tworzydlo J., Tajic A., Schomerus H., Beenakker C.W.: Dynamical model for the quantum-to-classical crossover of shot noise. Phys. Rev. B 68, 115313 (2003)CrossRefADSGoogle Scholar
  47. 47.
    Voros A.: Unstable periodic orbits and semiclassical quantisation. J. Phys. A 21, 685–692 (1988)CrossRefzbMATHADSMathSciNetGoogle Scholar
  48. 48.
    Wirzba A.: Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems. Phys. Rep. 309, 1–116 (1999)CrossRefADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2011

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Stéphane Nonnenmacher
    • 1
  • Johannes Sjöstrand
    • 2
  • Maciej Zworski
    • 3
    Email author
  1. 1.Institut de Physique Théorique, CEA/DSM/PhT, Unité de Recherche Associée au CNRS, CEA-SaclayGif-sur-YvetteFrance
  2. 2.Institut de Mathématiques de BourgogneUFR Science et TechniquesDijon CedexFrance
  3. 3.Mathematics DepartmentUniversity of CaliforniaBerkeleyUSA

Personalised recommendations