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A system of two bilinearly coupled harmonic oscillators has been solvedanalytically by using the Born-Oppenheimer (BO) product wavefunction ansatz andthe phase-space bound trajectory approach [J. S. Molano et al., Chem. Phys.Lett. \textbf{76}(12), 138171 (2021)]. The bilinearly coupled oscillator systemallows to obtain the analytical expression of the quantum system, facilitatingcomparison with the results of using the BO ansatz product. The analytical andBO wavefunctions are obtained as a product of parabolic cylinder functions. Thearguments of the parabolic cylinder functions of the exact and the BOwavefunctions are related to the physical coordinates by lineartransformations, represented by matrices $\mathsf{G}$ and $\mathsf{\tilde{G}}$respectively. A $QU$ decomposition of the matrix $\mathsf{G}$ outputs an uppertriangular matrix $\mathsf{U}$ that is closely related to the BO$\mathsf{\tilde{G}}$ matrix. The effect of the BO non-adiabatic coupling isanalyzed on the stability of the phase-space equations of motion. Theeigenvalues and eigenfunctions obtained by the Born-Oppenheimer phase-spacetrajectory approach show excellent agreement with the analytical solutions.

The use of the Born-Oppenheimer factorization in the phase-space  representation of the time-independent Schr\"odinger equation for bilinearly  coupled harmonic oscillators