Excitation of a quantum oscillator in thermal equilibrium: description with a spectral function

An approximate expression for the spectral function of a quantum oscillator in thermal equilibrium is derived, which describes the probability of energy transfer to the oscillator from an external force. The obtained expression coincides with the spectral shape of the photoabsorption cross-section line by electron centers in a solid in the presence of a strong electron-phonon bond. In the limiting cases of high and low temperatures, the dependences of the main characteristics of the energy transfer probability on the process parameters are analyzed.