We consider the problem of a single quantum oscillator coupled linearly to a heat bath of independent harmonic modes. An exact solution is presented for the system-oscillator observables of interest. The exact results are then used to evaluate the utility of a variational approach to the problem that has proven useful recently in elucidating the dynamics of dissipatively coupled systems. We find that the variational approach does provide a good description for most, but not all, observables of interest. Both the exact and the variational treatment demonstrate the important role played by the low-frequency bath modes in determining qualitative features of the dynamical behavior.
P. E. Parris and R. J. Silbey, "Variational Treatment of a Harmonic Oscillator Coupled to a Dissipative Heat Bath," Journal of Chemical Physics, vol. 86, no. 11, pp. 6381-6387, American Institute of Physics (AIP), Feb 1987.
The definitive version is available at https://doi.org/10.1063/1.452426
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