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arXiv · 1206.0716

Classical and Quantum Modes of Coupled Mathieu Equations

H. Landa·M. Drewsen·B. Reznik·A. Retzker
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Abstract

We expand the solutions of linearly coupled Mathieu equations in terms of infinite-continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the ('Floquet-Lyapunov') transformation to coordinates in which the motion is that of decoupled linear oscillators. We use this transformation to solve the Heisenberg equations of the corresponding quantum-mechanical problem, and find the quantum wavefunctions for stable oscillations, expressed in configuration-space. The obtained transformation and quantum solutions can be applied to more general linear systems with periodic coefficients (coupled Hill equations, periodically driven parametric oscillators), and to nonlinear systems as a starting point for convenient perturbative treatment of the nonlinearity.

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H. Landa, M. Drewsen, B. Reznik, A. Retzker. 2012-06-04. Classical and Quantum Modes of Coupled Mathieu Equations. https://doi.org/10.1088/1751-8113/45/45/455305

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