Towards Rigorous Derivation of Quantum Kinetic Equations

We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and by the Cauchy problem of the generalized quantum kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is established in the space of trace class operators. The links of the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.Comment: 25 page

The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems

The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear equations is constructed in the form of an expansion over particle clusters whose evolution is described by the corresponding order cumulant (semi-invariant) of evolution operators for the von Neumann equations. For the initial data from the space of sequences of trace class operators the existence of a strong and a weak solution of the Cauchy problem is proved. We discuss the relationships of this solution both with the $s$-particle statistical operators, which are solutions of the BBGKY hierarchy, and with the $s$-particle correlation operators of quantum systems.Comment: 26 page

On the solutions of the nonlinear Liouville hierarchy

We investigate the initial-value problem of the non-linear Liouville hierarchy. For the general form of the interaction potential we construct an explicit solution in terms of an expansion over particle clusters whose evolution is described by the corresponding-order cumulant of evolution operators of a system of finitely many particles. For the initial data from the space of integrable functions the existence of a strong solution of the Cauchy problem is proved.Comment: 9 page

On Rigorous Derivation of the Enskog Kinetic Equation

We develop a rigorous formalism for the description of the kinetic evolution of infinitely many hard spheres. On the basis of the kinetic cluster expansions of cumulants of groups of operators of finitely many hard spheres the nonlinear kinetic Enskog equation and its generalizations are justified. It is established that for initial states which are specified in terms of one-particle distribution functions the description of the evolution by the Cauchy problem of the BBGKY hierarchy and by the Cauchy problem of the generalized Enskog kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.Comment: 28 page

The spectral shift function and Levinson's theorem for quantum star graphs

We consider the Schr\"odinger operator on a star shaped graph with $n$ edges joined at a single vertex. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula

The Generalized Star Product and the Factorization of Scattering Matrices on Graphs

In this article we continue our analysis of Schr\"odinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discussed

Parents and Children: Way to Choice of Profession

It is reported the monograph “Parent assistance in professional self-determination of senior pupils” (2016), prepared by T. D. Zelenkina, E. Yu. Pryazhnikova and M. G. Sergeyeva, touches upon the problem of effective vocational guidance. The value of the edition is based on the fact that problems of choosing the profession do not lose its relevance. The views of the authors of the monograph on parental involvement in career choice of their children are characterized. For example, it is noted that the family may not only contribute to the process of professional self-determination, but also prevent correct choice. Model of training parents to help in professional self-determination of senior pupils proposed by the authors is briefly described

A New Look at Calcium Digermanide CaGe2_2: A High-Performing Semimetal Transparent Conducting Material for Ge Optoelectronics

Following a recently manifested guide of how to team up infrared transparency and high electrical conductivity within semimetal materials [C. Cui $et$ $al.$ Prog. Mater. Sci. 2023, 136, 101112], we evaluate an applicability of the calcium digermanide (CaGe$_2$) thin film electrodes for the advanced Ge-based optical devices. Rigorous growth experiments were conducted to define the optimal annealing treatment and thickness of the Ca-Ge mixture for producing stable CaGe$_2$ layers with high figure of merit (FOM) as transparent conducting material. Ab-initio electronic band structure calculations and optical modeling confirmed CaGe$_2$ semimetal nature, which is responsible for a demonstrated high FOM. To test CaGe$_2$ electrodes under actual conditions, a planar Ge photodetector (PD) with metal-semiconductor-metal structure was fabricated, where CaGe$_2$/Ge interface acts as Schottky barrier. The resulting Ge PD with semimetal electrodes outperformed commercially available Ge devices in terms of both photoresponse magnitude and operated spectral range. Moreover, by using femtosecond-laser projection lithography, a mesh CaGe$_2$ electrode with the relative broadband transmittance of 90\% and sheet resistance of 20 $\Omega$/sq. was demonstrated, which further enhanced Ge PD photoresponse. Thus, obtained results suggest that CaGe$_2$ thin films have a great potential in numerous applications promoting the era of advanced Ge optoelectronics.Comment: 12 pages, 4 figure