Yu. A. Kurochkin

Nonlinear symmetry algebra of the MIC-Kepler problem on the sphere S3.

The quantum mechanical problem of motion in the dual charged Coulomb field modified by a centrifugal term (MIC-Kepler problem) is considered in the three-dimensional space of constant positive curvature S3. Conserved operators in this problem form a cubic algebra similar to that of the Kepler problem on S3. The explicit form of invariants of this algebra shows that its representation associated with the MIC-Kepler problem on S3 is a nondegenerate one. The cubic symmetry algebra is used to obtain the energy spectrum of the problem.

Localization of Spherical Particles under the Action of Gradient Forces in the Field of a Zero-Order Bessel Beam. Rayleigh–Gans Approximation

The amplitude of the gradient force acting on a transparent spherical particle in the field of a zero-order Bessel beam has been calculated in the Rayleigh–Gans approximation. The expression obtained for the gradient-force amplitude takes into account the heterogeneity of the acting radiation in the volume of the particle. The optimal conditions of trapping and transportation of the particle (parameters of the particle, liquid, and of the Bessel beam) to the localization region have been determined using the solution of the kinetic equation of particle motion in a liquid. It is shown that for certain relationships between the particle radius and the Bessel beam width the localization region is shifted relative to the central maximum of the beam. This is due to the equal action of the gradient forces caused by the central maximum and the first interference ring of the Bessel beam. A qualitative comparison of the results obtained with the known experimental data has been performed.

On some integrable systems in the extended lobachevsky space

Some classical and quantum-mechanical problems previously studied in Lobachevsky space are generalized to the extended Lobachevsky space (unification of the real, imaginary Lobachevsky spaces and absolute). Solutions of the Schrödinger equation with Coulomb potential in two coordinate systems of the imaginary Lobachevsky space are considered. The problem of motion of a charged particle in the homogeneous magnetic field in the imaginary Lobachevsky space is treated both classically and quantum mechanically. In the classical case, Hamilton-Jacoby equation is solved by separation of variables, and constraints for integrals of motion are derived. In the quantum case, solutions of Klein-Fock-Gordon equation are found.

Vector parameterization of the Lorentz group transformations and polar decomposition of Mueller matrices

It is shown that the representation of the coherence matrix (the polarization density matrix) of beams of electromagnetic waves as a biquaternion corresponding to the four-vector of a pseudo-Euclidean space whose components are the intensity and the Stokes parameters provides a possibility of introducing the group transformations of these quantities isomorphic to SO(3.1) group. These transformations are a subset of the set of Mueller polarization matrices which, generally speaking, form a semigroup. The reduction of the semigroup of Mueller matrices to the group of transformations opens the possibility to use the vector parameterization of SO(3.1) group for interpretation of the polar decomposition of Mueller matrices. In particular, in this approach, the elements of the Mueller matrices corresponding to phase elements and polarizers turn out to be most simply and naturally related to their eigenpolarizations.

The Higgs algebra and the Kepler problem in R3

The quantum mechanical correspondence of the Kepler problem in R3 and the free-particle motion on spaces S3 and S31 is found by using the fact that the Higgs algebra is a finite W-algebra obtained by embedding of algebra sl(2) into sl(4).

Motion Caused by Magnetic Field in Lobachevsky Space

We study motion of a relativistic particle in the 3‐dimensional Lobachevsky space in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in the Euclidean space. Three integrals of motion are found and equations of motion are solved exactly in the special cylindrical coordinates. Motion on surface of the cylinder of constant radius is considered in detail.

An algebraic treatment of the MIC-Kepler problem on S3 sphere

The quantum-mechanical problem of motion in a dual charged Coulomb field modified by a centrifugal term (MIC-Kepler problem) is considered in a three-dimensional space of constant positive curvature, S3. Conserved operators are found, and their commutation relations are derived. It is shown that, in the MIC-Kepler problem in S3 space, conserved operators form a cubic algebra similar to that of the Kepler problem in the same space. This symmetry algebra is used to obtain the energy spectrum of the problem.

Regge trajectories of the Coulomb potential in the space of constant negative curvature S31

Analytic properties of the scattering amplitude for Coulomb potential on the background of the space of constant negative curvature are studied. Special attention is given to the comparison of the Regge trajectories for curved and flat spaces. We show that there exist considerably differences in the behavior of the Regge trajectories in these spaces.

Kepler motion on single-sheet hyperboloid

The classical Kepler–Coulomb problem on the single-sheeted hyperboloid H31 is solved in the framework of the Hamilton–Jacobi equation. We have proven that all the bounded orbits are closed and periodic. The paths are ellipses or circles for finite motion.

Magnetic field in the Lobachevsky space and related integrable systems

Various possibilities to define analogs of the uniform magnetic field in the Lobachevsky space are considered using different coordinate systems in this space. Quantum mechanical problem of motion in the defined fields is also treated. Variables in the Schrödinger equation are separated and diagonal operators are found. For some cases, exact solutions are obtained.