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

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...

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Bibliographic Details
Published in:Theoretical and mathematical physics 2019, Vol.198 (1), p.17-28
Main Author: Chistyakov, V. V.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Bifurcations
Collapse
Initial conditions
Mathematical analysis
Mathematical and Computational Physics
Physics
Physics and Astronomy
Schrodinger equation
Spreading
Theoretical
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Summary: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.
ISSN:0040-5779
1573-9333
DOI:10.1134/S0040577919010021