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

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

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)

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

Analysis of spatial humus distribution in virgin and arable Chernozems in the eroding landscapes

A statistical assessment of data on the content and spatial distribution of humus in virgin and arable Chernozems on slopes in the south of the forest-steppe zone of the Central Russian Upland was performed. The greatest number of statistically significant differences between arable and virgin Chernozems was found for the upper soil layer (0–20 cm

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

Individual personality traits as predictors of intra-organizational vertical career growth of employees

The paper is devoted to the research on the influence of individual personality traits on the success of intra-organizational vertical career growth of employees, the indicator of which is the promotion of employees in their position (status) in the organization, taking into account their speed of passing the career levels. The article analyzes the types of career, criteria and factors of career growth, including personality characteristics in the context of their influence on career processes. The article presents the results of an empirical study of the managers of JSC Concern Energomera, Stavropol (Russia), which determined the relationship between the success rate of intra-organizational vertical career growth and individual personality characteristics, manifested in the level of mental tension (anxiety, suspicion, inclination to risk, spontaneity and emotional sensitivity) and expressiveness. The authors did not find statistically significant contribution of regulatory properties and indicators of general mental abilities to distinguishing groups of employees with high and low indicators of vertical career growth. The averaged personal profiles of employees with high and low indicators of vertical career growth are presented.peer-reviewe