Geodesic
Mathieu quantum oscillator with cubic force, friction and noise
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 24 (2016) no. 3, pp. 54-67.

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Mathieu quantum oscillator as a generalization of the classical one with cubic force, friction and noise has been proposed. The problem of the transition from classical to quantum behavior is of importance not only for fundamental knowledge but also for applications. As an example one can mention the oscillatory motion of low mass objects at decreasing temperature. The equation describing Mathieu quantum oscillator is Schroedinger–Lanfevin–Kostin equation with the quartic potential, logarithmic dissipative and Langevin terms. The numerical integration of this equation was performed at specified initial and boundary conditions by using the iterative finite-difference method. At weak parametric external action, one or two spectral components on transition frequencies are generated as well as components on combined frequencies caused by the Floquet states. If the parametric amplitude increases then the number of spectral components is also grown. The friction causes damping as in the classic Mathieu oscillator. Multi-frequency regime of Mathieu quantum oscillator occurs if the parametric action amplitude is equal to or exceeds unity wherein the parametric frequency is not coupled rationally with spectrum frequencies. The frequency differences of neighboring spectral components can be equal to one of two frequencies in spectrum or its combination. The Gaussian white noise changes the realizations: at small friction coefficient and moderate noise intensity, the spectral components on combined frequencies become hidden, only component on transition frequency from ground state into first excited remains noticeable. Thus, the research shows a significant dependence of the oscillations on the model parameters. Increasing the amplitude of the external action leads to complication of the spectra.
Keywords: Quantum Mathieu oscillator, cubic force, quasiperiodicity, noise excitation. 
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     author = {A. L. Sanin and A. A. Smirnovsky},
     title = {Mathieu quantum oscillator with cubic force, friction and noise},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {54--67},
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     year = {2016},
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     url = {http://geodesic.mathdoc.fr/item/IVP_2016_24_3_a4/}
}
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A. L. Sanin; A. A. Smirnovsky. Mathieu quantum oscillator with cubic force, friction and noise. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 24 (2016) no. 3, pp. 54-67. http://geodesic.mathdoc.fr/item/IVP_2016_24_3_a4/