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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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We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameter h 1. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to an effective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.