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The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ( → 0), are constructed with a power accuracy of O( N/2 ), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.2000 Mathematics Subject Classification: 81Q20, 35Q55, 47G20.

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We study linear problems of mathematical physics in which the adiabatic approximation is used in the wide sense. Using the idea that all these problems can be treated as problems with operator-valued symbol, we propose a general regular scheme of adiabatic approximation based on operator methods. 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, subregions of dimensional quantization, etc.) with the possible degeneration taken into account. Next, applying the asymptotic methods in particular, the semiclassical approximation method, to the reduced equation, one can classify the states corresponding to a distinguished term (effective Hamiltonian). We show that the adiabatic effective Hamiltonian and the semiclassical Hamiltonian can be different, which results in the appearance of "nonstandard characteristics" while one passes to classical mechanics. This approach is used to construct solutions of several problems in wave and quantum mechanics, in particular, problems in molecular physics, solid-state physics, nanophysics, hydrodynamics.

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