The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration

Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operato

The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k)

ORGANIZATIONAL AND LEGAL REGULATION PROCEDURE FOR CIRCULATION OF EXTEMPORAL MEDICINES BASED ON PHARMACEUTICAL LAW

Introduction. The paper studied the situation regarding the production of medicines in pharmacies. Established that the presence of the pharmacy manufacture of medicines, patients entitled to receive medicines made to the needs of individuals. Goal – to study the organizational and legal procedure of regulation of extemporal medicines by developing of the algorithm for determining the legal act used in the event of conflict based on the law pharmaceutical law. Materials and methods. The materials of the study were legal acts of Ukraine: Laws of Ukraine, Decrees of the Cabinet of Ministers of Ukraine, Orders of the Ministry of Healthcare of Ukraine. The research methods were legal, documentary and comparative analyzes. Results and discussion. However, production of extemporaneous preparations in the pharmacy requires a production base and the appropriate staff. Therefore, the authors based on pharmaceutical law proposed organizational and legal procedure regarding the regulation of extemporaneous preparations by developing the algorithm for determining the legal act which should follow in the event of conflicts concerning the law. Conclusions. Based on pharmaceutical law held organizational and legal procedure for the regulation of circulation of extemporal medicines. Proposed the algorithm for determining of the legal act used in the practice of pharmacy professionals. Considered the professional status of an authorized person on the stage of quality control of extemporal medicines in their treatment in healthcare institutions of private property

Symmetry operators and separation of variables in the (2+1)(2+1)-dimensional Dirac equation with external electromagnetic field

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski (flat) space. For each of the sets we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.Comment: 24 pages, version accepted for publication in Int. J. Geom. Methods Mod. Phy

Harmonic oscillator coherent states from the orbit theory standpoint

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schr\"odinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution of a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.Comment: 16 page

On the new approach to variable separation in the time-dependent Schr\"odinger equation with two space dimensions

We suggest an effective approach to separation of variables in the Schr\"odinger equation with two space variables. Using it we classify inequivalent potentials $V(x_1,x_2)$ such that the corresponding Schr\" odinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schr\" odinger equation with the anisotropic harmonic oscillator potential $V=k_1x_1^2+k_2x_2^2$ and obtain a complete list of coordinate systems providing its separability. Most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free Schr\" odinger equation ($V=0$).Comment: 21 pages, latex, to appear in the "Journal of Mathematical Physics" (1995

Noncommutative reduction of nonlinear Schrödinger equation on Lie groups

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration to the linear part of a nonlinear equation, which allows one to find bases in the space of solutions of linear partial differential equations with a set of noncommuting symmetry operators. The approach is implemented for the generalized nonlinear Schrödinger equation on a Lie group in curved space with local cubic nonlinearity. General formalism is illustrated by the example of the noncommutative reduction of the nonstationary nonlinear Schrödinger equation on the motion group E(2) of the two-dimensional plane R2. In this particular case, we come to the usual (1+1)-dimensional nonlinear Schrödinger equation with the soliton solution. Another example provides the noncommutative reduction of the stationary multidimensional nonlinear Schrödinger equation on the four-dimensional exponential solvable group

Possibilities of laser spectroscopy for monitoring the profile dynamics of the volatile metabolite in exhaled air

In this work we studied applicability of the laser spectroscopy for fixing differences in composition of exhaled air depending on the position of the body in different physical states. Using principal component analysis we show that the use of the laser spectroscopy methods is sufficiently effective to solve this problem and provide additional opportunities for the comprehensive study of the human condition