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Библиогр.: 9 назв.

The Dirac equation for a charge in an external electromagnetic field is the basic equation for
relativistic quantum mechanics and quantum electrodynamics. The Dirac equation in the magnetic solenoid field
is the basis of the theory of Aharonov-Bohm effect both in (3+1) and (2+1) dimensions. The self-adjoint
extension problem in quantum mechanics was investigated in detail by Gitman, Tyutin, and Voronov. The Dirac
equation is of interest in planar gravity and the Bañados-Teitelboim-Zanelli (BTZ) black hole in relation with
the investigation of the Dirac matter field behavior on the background of the BTZ gravity. Another motivation in
studying the (2 + 1) Dirac equation in curved space-time is that, although the (2+1) dimensional gravity is a toy
model for a regular Einstein theory in (3+1) dimensions, it preserves some significant properties of regular
gravity being mathematically simpler. To construct exact solutions of the Dirac equation, the separation of
variables method is commonly used. A new method, named the noncommutative integration method, has been
proposed to construct basses of exact solutions of linear partial differential equations. The noncommutative
integration (NI) method essentially uses a Lie algebra L of differential symmetry operators of the first order.
Note that the method allows one to find exact solutions (NI-solutions) in the cases when the Dirac equation does
not allow separation of variables.