Superoperator method in the theory of an oscillator which interacts weakly with a medium. I. General results

The superoperator method is used to construct the quantum theory of an oscillator which interacts weakly with a medium. General expressions are obtained for the spectral representations of the correlations functions of the displacements and the population numbers of the oscillator, and also kinetic equations for the time correlation functions. These results make it possible to investigate the dynamics of an oscillator when allowance is made for any power of the oscillator displacement operator in the Hamiltonian describing the nonlinearity of the oscillator and in the Hamiltonian describing the interaction between the oscillator and the fluctuations of the medium. The influence of fluctuations of the medium on the dynamics of the oscillator is described by the correlation functions for the potential energy of the medium. This yields, in particular, the fine structure in the spectral distribution of a nonlinear oscillator, the line widths of the structure being expressed in terms of the correlation function of the medium. In the theory developed here, the medium with which the oscillator interacts is not particularized.