Classical and Quantum Modes of Coupled Mathieu Equations

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.