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.