Operator separation of variables for adiabatic problems in quantum and wave mechanics

Linear problems in mathematical physics where the adiabatic approximation is used in a wide sense are studied. From the idea that all these problems can be treated as problems with an operator-valued symbol, a general regular scheme of adiabatic approximation based on operator methods is proposed. This scheme is a generalization of the Born-Oppenheimer and Maslov methods, the Peierls substitution, etc. The approach proposed in this paper allows one to obtain "effective" reduced equations for a wide class of states inside terms (i.e., inside modes, subbands of dimensional quantization, etc.) with possible degeneration taken into account. Next, by application of asymptotic methods, in particular the semiclassical approximation method, to the reduced equation, the states corresponding to a distinguished term (effective Hamiltonian) can be classified. It is shown that the adiabatic effective Hamiltonian and the semiclassical Hamiltonian can be different, which results in the appearance of "nonstandard characteristics" while passing to classical mechanics. This approach is used to construct solutions of several problems in wave and quantum mechanics, particularly problems in molecular physics, solid-state physics, nanophysics and hydrodynamics.