I. V. Shirokov

INTEGRATION OF THE DIRAC EQUATION, WHICH DOES NOT PRESUME COMPLETE SEPARATION OF VARIABLES, IN STACKEL SPACES

The method of noncommutative integration of linear differential equations is used to construct an exact solution of the Dirac equation, which does not presume complete separation of variables, in Stäckel spaces. The Dirac equation in an external electromagnetic field is integrated by this method, using one example. The Stäckel space under consideration does not enable one to solve this equation exactly within the framework of the theory of separation of variables.

Application of the K-Orbit Method to a Solution of Thermodynamics Problems for Noncompact Lie Groups

A method of solving the main thermodynamics problem of homogeneous spaces for noncompact manifolds (on the example of the noncompact unimodular Lie groups) is suggested in the present paper. The method is based on the K-orbit formalism. A formula is derived that allows the statistical sum in noncompact spaces and the Greens function of a scalar particle in background gravitational field to be calculated. The method is illustrated by an example.

Four-dimensional Lie group integration of the Klein-Fock equation

A method of four-dimensional Lie group integration of the Fock–Klein equation is described in the present paper. The method is based on the formalism of co-adjoint representation orbits and on the non-commutative integration method. Right-invariant metrics being solutions of the Einstein equation are also classified for manifolds of four-dimensional Lie groups.

The K-Orbits, Identities, and Classification of Four-Dimensional Homogeneous Spaces with the Group of Poincaré and de Sitter Transforms

The method of orbits traditionally applied to geometric quantization problems is used to study homogeneous spaces. Based on the proposed classification of the orbits of co-adjoint representation (K-orbits), a classification of homogeneous spaces is constructed. This classification allows one, in particular, to point out the explicit form of identities – functional relations between the transform-group generators – which are of great importance in applied problems (e.g., in the theory of separation of variables). All four-dimensional homogeneous spaces with the group of Poincaré and de Sitter transforms are classified and all independent identities on these spaces are given in explicit form.

Classification of quadratic symmetry algebras of the Schrdinger equation

Noncommuntative quadratic symmetry algebras of a certain class for the Schrödinger equation are classified. For each such algebra, the permissible potential is found. The application of noncommuntative integration of partial differential equations by means of quadratic algebras is demonstrated for a nontrivial example. The solution obtained forms the basis for the representation of quadratic algebras.