Skip to main content
SpringerLink
Log in

Quasi-energies and chaotic behaviour of a periodically delta-kicked quantum singular oscillator

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

The interrelations between the evolution operator, monodromy matrix, and quasi-energy spectrum of the time-dependent «singular oscillator» affected by periodic perturbations of the coefficients of the Hamiltonian which is a linear combination ofSU(1, 1)-group generators are established both in adiabatic and «delta-kicked» regimes. Universal invariants conserved in time independently of the concrete values of the coefficients are constructed. The connection between the concepts of «quantum chaos» and «quasi-energy spectra» is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Gasati, B. V. Chirikov, D. L. Shepelynsky andI. Guaener:Phys. Rep.,154, 77 (1987).

    Article  ADS  Google Scholar 

  2. J. Bellisard: inTrends and Developments in the Eighties, edited byS. Albeverio andPh. Blanchard (World Scientific, Singapore, 1985).

    Google Scholar 

  3. G. Karner, V. I. Man’ko andL. Streit: inIntegrals of Motion and Theory of Interaction of Multilevel Systems with Quantized Fields, Proceedings of Lebedev Physics Institute, edited byM. A. Markov andV. I. Man’ko, Vol.208 (Nauka, Moscow, 1992), p. 226 (English translation by Nova Science, Commack, N.Y., 1993);Rep. Math. Phys.,27, 177 (1991).

    Google Scholar 

  4. P. Camiz, A. Gerardi, C. Marchioro, E. Presutti andE. Scacciatelli:J. Math. Phys.,12, 2040 (1971).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. V. V. Dodonov, I. A. Malkin andV. I. Man’ko:Physica,72, 597 (1974).

    Article  MathSciNet  ADS  Google Scholar 

  6. V. V. Dodonov, I. A. Malkin andV. I. Man’ko:Nuovo Cimento B,24, 46 (1974).

    Article  MathSciNet  ADS  Google Scholar 

  7. D. Peak andA. Inomata:J. Math. Phys.,10, 1422 (1969).

    Article  MathSciNet  ADS  Google Scholar 

  8. V. V. Dodonov, I. A. Malkin andV. I. Man’ko:Phys. Lett. A,39, 377 (1972).

    Article  MathSciNet  ADS  Google Scholar 

  9. F. Calogero:J. Math. Phys.,10, 2191 (1969);12, 419 (1971);Lett. Nuovo Cimento,20, 251 (1977).

    Article  MathSciNet  ADS  Google Scholar 

  10. A. V. Gorokhov:Izv. Vyssh. Uchebn. Zaved., Fiz.,2, 148 (1971).

    Google Scholar 

  11. H. Hartmann:Theor. Chim. Acta (Berlin),24, 201 (1972).

    Article  Google Scholar 

  12. A. Hautot:J. Math. Phys.,14, 1320 (1973).

    Article  ADS  Google Scholar 

  13. V. de Alfano, S. Fubini andG. Furlan:Nuovo Cimento A,34, 569 (1976).

    Article  ADS  Google Scholar 

  14. M. M. Nieto:Phys. Rev. D,20, 1332 (1979);M. M. Nieto andL. M. Simmons jr.:Phys. Rev. D,22, 391 (1980);V. P. Gutschick, M. M. Nieto andL. M. Simmons jr.:Phys. Lett. A,76, 15 (1980).

    Article  MathSciNet  ADS  Google Scholar 

  15. J. G. Hartly andJ. K. Ray:Phys. Rev. A,24, 2873 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  16. V. V. Dodonov andV. I. Man’ko: inGroup Theory, Gravitation and Elementary Particle Physics, Proceedings of Lebedev Physics Institute, edited byA. A. Komar, Vol.167 (Nauka, Moscow, 1986), p. 7 (English translation by Nova Science, Commack, N.Y., 1987).

    Google Scholar 

  17. C. Quesne:J. Phys. A,21, 3093 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  18. J. M. Cai andA. Inomata:Phys. Lett. A,141, 315 (1989).

    Article  MathSciNet  ADS  Google Scholar 

  19. A. Barut andL. Gieardello:Commun. Math. Phys.,21, 41 (1971).

    Article  ADS  MATH  Google Scholar 

  20. E. Onofri andM. Pauri:Lett. Nuovo Cimenta,3, 35 (1972).

    Article  MathSciNet  Google Scholar 

  21. C. C. Gerry andS. Silverman:J. Math. Phys.,23, 1995 (1982);Phys. Rev. A,29, 1574 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  22. C. C. Gerry:J. Phys. A,16, LI (1983);Phys. Rev. A,33, 2207 (1986).

    Article  MathSciNet  Google Scholar 

  23. I. H. Duru:Phys. Rev. D,30, 2121 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. S. M. Chumakov, V. V. Dodonov andV. I. Man’ko:J. Phys. A,19, 3229 (1986); inGroup Theory, Gravitation and Elementary Particle Physics, Proceedings of Lebedev Physics Institute, edited byA. A. Komar, Vol.167 (Nauka, Moscow, 1986), p. 162 (English translation by Nova Science, Commack, N.Y., 1987, p. 209).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. G. Datolli, A. Torre andR. Caloi:Phys. Rev. A,33, 2789 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  26. G. Datolli, A. Dipace andA. Torre:Phys. Rev. A,33, 4387 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  27. G. Datolli, S. Solimeno andA. Torre:Phys. Rev. A,34, 2646 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  28. M. Bohm andG. Junker:J. Math. Phys.,28, 1978 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  29. A. O. Barut, A. Inomata andG. Junker:J. Phys. A,23, 1179 (1990).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. A. O. Barut andI. H. Duru:Phys. Lett. A,158, 441 (1991).

    Article  ADS  Google Scholar 

  31. M. V. Berry:Proc. R. Soc. London, Ser. A,392, 45 (1984).

    Article  ADS  Google Scholar 

  32. I. A. Malkin andV. I. Man’ko: inDynamical Symmetries and Coherent States of the Quantum Systems (Nauka, Moscow, 1979), p. 320 (in Russian).

    Google Scholar 

  33. V. V. Dodonov, A. B. Klimov andV. I. Man’ko: inSqueezed and Correlated States of Quantum Systems, Proceedings of Lebedev Physics Institute, edited byA. A. Markov, Vol.200 (Nauka, Moscow, 1991), p. 56 (English translation by Nova Science, Commack, N.Y., 1992).

    Google Scholar 

  34. I. A. Malkin, V. I. Man’ko andD. A. Trifonov:J. Math. Phys.,14, 576 (1973).

    Article  ADS  Google Scholar 

  35. M. H. Engineer andG. Ghosh:J. Phys. A,21, L95 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  36. V. V. Dodonov andV. I. Man’ko: inGroup Theoretical Methods in Physics, Proceedings of the Second International Seminar at Zvenigorod, November 1982, edited byA. A. Markov, V. I. Man’ko andA. E. Shabad, Vol. 1 (Harwood Academic Publishers, 1985), p. 591.

  37. V. V. Dodonov andV. I. Man’ko: inInvariants and the Evolution of Nonstationary Quantum Systems, Proceedings of Lebedev Physics Institute (Nauka, Moscow, 1987), Vol.183 (English translation by Nova Science, Commack, N.Y., 1989).

    Google Scholar 

  38. V. I. Man’ko:Geometrical phase, generalised quasienergy and Floquet operator as invariants, inProceedings of the V Conference on Symmetry in Physics, Bregenz, August 1992, edited byB. Gruber (Plenum Press, 1993).

Download references

Author information

Authors and Affiliations

  1. Lebedev Physics Institute, Leninsky Prospect 53, 117924, Moscow, Russia

    V. V. Dodonov, V. I. Man’ko & D. V. Zhivotchenko

Authors
  1. V. V. Dodonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. I. Man’ko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. V. Zhivotchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The authors of this-paper have agreed to not receive the proofs for correction.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dodonov, V.V., Man’ko, V.I. & Zhivotchenko, D.V. Quasi-energies and chaotic behaviour of a periodically delta-kicked quantum singular oscillator. Nuov Cim B 108, 1349–1363 (1993). https://doi.org/10.1007/BF02755189

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02755189

PACS

PACS

Search

Navigation