Abstract
In this paper, we establish that in time-dependent harmonic oscillator under an oscillatory decreasing perturbation the parametric resonances, which do not exist in an ordinary harmonic oscillator under the same perturbation, may occur.
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Bogoliubov N.N., Mitropolskiy Yu.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York (1961)
Broer H.W.: Resonance and fractal geometry. Acta Appl. Math. 120(1), 61–86 (2012)
Burd, V.: Method of averaging for differential equations on an infinite interval: theory and applications. In: Lecture Notes in Pure and Applied Mathematics, vol. 255. Chapman & Hall/CRC, Boca Raton (2007)
Burd V.Sh., Karakulin V.A.: On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients. Math. Notes 64(5), 571–578 (1998)
Burd V., Nesterov P.: Parametric resonance in adiabatic oscillators. Results Math. 58, 1–15 (2010)
Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
Denisov, S.A., Kiselev, A.: Spectral properties of Schrödinger operators with decaying potentials. In: Simon Festschrift B (ed.) Proceedings of Symposia in Pure Mathematics. vol. 76, part 2. American Mathematical Society, Providence, pp. 565–589 (2007)
Eastham, M.S.P.: The asymptotic solution of linear differential systems. London Mathematical Society Monographs. Clarendon Press, Oxford (1989)
Harris, W.A. Jr., Lutz, D.A.: On the asymptotic integration of linear differential systems. J. Math. Anal. Appl. 48(1), 1–16 (1974)
Harris W.A. Jr, Lutz D.A.: A unified theory of asymptotic integration. J. Math. Anal. Appl. 57(3), 571–586 (1977)
Levinson N.: The asymptotic nature of the solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)
Lukic, M.: Schrödinger operators with slowly decaying Wigner-von Neumann type potentials. J. Spectr. Theory 3(2), 147–169 (2013)
Naboko S., Simonov S.: Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential. Math. Proc. Cambridge Philos. Soc. 153(1), 33–58 (2012)
Nesterov P.N.: Construction of the asymptotics of the solutions of the one-dimensional Schrödinger equation with rapidly oscillating potential. Math. Notes 80(2), 233–243 (2006)
Nesterov P.N.: Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients. Differ. Equ. 43(6), 745–756 (2007)
Nesterov P.: On eigenvalues of the one-dimensional Dirac operator with oscillatory decreasing potential. Math. Phys. Anal. Geom. 15(3), 257–298 (2012)
Perelomov A.: Generalized Coherent States and Their Applications. Springer, Berlin (1986)
Sagdeev R.Z., Usikov D.A., Zaslavsky G.M.: Nonlinear Physics: From Pendulum to Turbulence and Chaos. Harwood Academic Publishers, New York (1988)
Turner K.L., Miller S.A., Hartwell P.G., MacDonald N.C., Strogatz S.H., Adams S.G.: Five parametric resonances in a microelectromechanical system. Nature 396, 149–152 (1998)
Um C.I., Yeon K.H., George T.F.: The quantum damped harmonic oscillator. Phys. Rep. 362(2-3), 63–192 (2002)
Wintner A.: The adiabatic linear oscillator. Am. J. Math. 68, 385–397 (1946)
Wintner A.: Asymptotic integration of the adiabatic oscillator. Am. J. Math. 69, 251–272 (1946)
Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients, vol. 1 and 2. Keter Publishing House, Jerusalem (1975)
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Nesterov, P. Appearance of New Parametric Resonances in Time-Dependent Harmonic Oscillator. Results. Math. 64, 229–251 (2013). https://doi.org/10.1007/s00025-013-0311-0
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DOI: https://doi.org/10.1007/s00025-013-0311-0