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

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