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


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.


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


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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

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

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