In the period from 1990 to 1994 the investigations of some mathematical models of heat processing of a steel were carried out. The solvability and the uniquiness of the boundary value problem for the system of integro-differential equations describing the austenite-perlite transformation under cooling were proved under different restrictions. The inverse problem of determination functional coefficients for this transformation on the base of temperature measurements were solved. In the period from 1995 to 2001 the investigations of the classical and quantum two-body problem with a central interaction on the complete two point homogineous Riemannian spaces (particulary on the spaces of a constant sectional curvature) were carried out. For the spaces of a constant sectional curvature the Hamiltonian reduction of the classical problem was carried out and the classification of the reduced dynamical systems with two degrees of freedom was given. For some values of the momentum map the conditions for the interaction were found, which provide the absence of particles collisions on the infinite period of time. In the quantum case for an arbitrary complete two point homogeneous Riemannian space the expression of Hamiltonian through the radial differential operator and generators of the symmetry group was found. This expression gives the possibilities to construct the selfadjoint extension of the Hamiltonian and to find some infinite eigenvalue series using the representation theory of groups. There was found the description of the reduced cotangent bundle of a homogeneous space of arbitrary Lie group in terms of orbits of the coadjoint action of that group. Noncomutative algebras of invariant differential operators on homogeneous spaces $U_{\mathbb{H}}(n+1)/(U_{\mathbb{H}}(n-1)U_{\mathbb{H}}(1)),\;U(n+1)/(U(n-1)U(1))$ and their noncompact analogues, connected with the quantum two body problem on spaces $P^n(\mathbb{H}),\; P^n(\mathbb{C}),\; H^n(\mathbb{H})$ and $H^n(\mathbb{C})$ were investigated.
Main publications:
Shchepetilov A. V. Reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature // J. Phys. A, 1998, 31, 6279–6291.
