Investigation of systems with quadratic interaction by the method of functional derivatives

For a fermion-boson system with quadratic interaction in the boson field in which the transition matrix element depends on the boson variables, a method is proposed for calculating the many-time statistical Green's functions on the basis of the method of functional derivatives. In the limit of a fixed fermion, the explicit form of the exact many-time Green's functions is found. For the case of a moving fermion, an iterative procedure is proposed for calculating the Green's functions with allowance for indirect interaction in the boson subsystem.