Advertisement

Theoretical and Mathematical Physics

, Volume 198, Issue 1, pp 17–28 | Cite as

Quantum Analogue of Unstable Limit Cycles of a Periodically Perturbed Inverted Oscillator

  • V. V. ChistyakovEmail author
Article
  • 7 Downloads

Abstract

To study the quantum analogue of classical limit cycles, we consider the behavior of a particle in a negative quadratic potential perturbed by a sinusoidal field. We propose a type of wave function asymptotically satisfying the operator of initial conditions and still admitting analytic integration of the nonstationary Schrödinger equation. The solution demonstrates that for certain perturbation phases determined by the forcing frequency and the initial indeterminacy of the coordinate, the wave-packet center temporarily stabilizes near the potential maximum for approximately two “natural periods” of the oscillator and then moves to infinity with bifurcations in the drift direction. The effect is not masked by packet spreading, because the packet undergoes anomalous narrowing (collapse) to a size of the order of the characteristic length on the above time interval and its unbounded spreading begins only after this.

Keywords

inverted quantum oscillator periodic perturbation limit cycle nonstationary Schrödinger equation generalized Gaussian type collapse dynamical stabilization bifurcation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Barton, “Quantum mechanics of the inverted oscillator potential,” Ann. Phys., 166, 322–363 (1986).ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Baskoutas, A. Jannussistl, and R. Mignanig, “Dissipative tunnelling of the inverted Caldirola–Kanai oscillator,” J. Phys. A: Math. Gen., 27, 2189–2196 (1994).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Sh. Matsumoto and M. Yoshimura, “Dynamics of barrier penetration in thermal medium: Exact result for inverted harmonic oscillator,” Phys. Rev. A, 63, 012104 (2000).ADSCrossRefGoogle Scholar
  4. 4.
    B. N. Zakhariev, “Discrete and continuous quantum mechanics, exactly solvable models (Lessons of quantum intuition II) [in Russian],” PEPAN, 23, 1387–1468 (1992).Google Scholar
  5. 5.
    C. A. Mu˜noz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A: Math. Theor., 42, 485210 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Maamache and J. R. Choi, “Quantum-classical correspondence for the inverted oscillator,” Chinese Phys. C, 41, 113106 (2017).ADSCrossRefGoogle Scholar
  7. 7.
    P. Duclosi, E. Soccorsi, P. Štovíček, and M. Vittot, “On the stability of periodically time-dependent quantum systems,” Rev. Math. Phys., 20, 725–764 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Y. Nogami and F. M. Toyama, “Nonlinear Schrödinger soliton in a time-dependent quadratic potential,” Phys. Rev. E, 49, 4497–4501 (1994).ADSCrossRefGoogle Scholar
  9. 9.
    G.-J. Guo, Z.-Z. Ren, G.-X. Ju, and X.-Y. Guo, “Quantum tunneling effect of a driven inverted harmonic oscillator,” J. Phys. A: Math. Theor., 44, 305301 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. G. Bagrov, D. M. Gitman, E. S. Macedo, and A. S. Pereira, “Coherent states of inverse oscillators and related problems,” J. Phys. A: Math. Theor., 46, 325305 (2013).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg National Research University of Information Technologies, Mechanics, and OpticsSt. PetersburgRussia

Personalised recommendations