Classical and Quantum Modes of Coupled Mathieu Equations

TL;DR

This paper develops a method to diagonalize coupled Mathieu equations using infinite-continued matrix inversions, enabling explicit solutions for classical modes and quantum wavefunctions in stable oscillations, with applications to broader linear and nonlinear systems.

Contribution

It introduces a novel approach to solve coupled Mathieu equations and derive explicit quantum solutions, extending to general linear systems with periodic coefficients.

Findings

Explicit diagonalization of coupled Mathieu equations

Quantum wavefunctions for stable oscillations derived

Applicable to general linear systems with periodic coefficients

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.