Operator method for coupled anharmonic oscillators

Two coupled anharmonic oscillators are considered as a model for a nonpertu rbative description of the correlation and nonadiabatic effects which are t ypical for many-dimensional quantum systems. The eigenvalues and eigenfunct ions for this model are found by means of the operator method, modified for the case of degenerate solutions of the Schrodinger equation. It is shown that the zeroth approximation of the method allows one to find the analytic al and uniformly suitable approximation for the energy levels and their spl itting in the entire range of Hamiltonian parameters and quantum numbers. N umerical calculations demonstrate the convergence of the successive approxi mations, even for quasistationary stales of the system. The results are of interest for applied problems of spectroscopy and solid state physics.