#### Goal of the project

The goal of the first part of the work is to study new quantum phenomena in micro - and nanostructures, interesting from a fundamental point of view and promising for applications. In particular, the study of the possibility of implementing quantum data transfer protocols on modern quantum computers, modeling of nontrivial effects of the interaction of matter with radiation by an example of superconducting systems, analysis of superfluid and luminescent properties of dipole exciton systems under Bose condensation and Cooper pairing, study of electronic and phonon properties of layered electrides, study of magnetoplasmons on the edges of graphene and analysis of the virial theorem and its relationship with the quantum pressure of electronic gas in graphene and other Dirac materials.

The goal of the second part of the work is to study the spectrum of the magnetic Schroedinger operator for a resonance quantum system with a small symmetry violation, in particular, the study of the spectrum of a particle in a strong magnetic field and in an electric field periodic in one direction and growing in the other direction. It is planned to construct a semiclassical asymptotics of spectral zones arising in splitting of Landau levels, to investigate an effective quantum operator of a two-dimensional Hamiltonian of a charge in a resonance electromagnetic trap, which asymptotically describes the anharmonic part of the Hamiltonian. To construct a semiclassical asymptotics of the spectrum and stationary states of this operator taking into account the description of exponentially small tunneling effects.

The goal of the third part of the work is to study the quantum three-frequency hyperbolic resonance oscillator and its symmetry algebra. It is planned to construct a finite set of generators of the algebra of symmetries with polynomial commutation relations, to construct irreducible representations of this algebra and the corresponding coherent states. This task is related to the objects of study in the first and second sections.

The goal of the fourth part of the work is to obtain an exact expression for the anomalous magnetic moment of an electron in a topologically massive two-dimensional electrodynamics in a constant magnetic field in the one-loop approximation and to study its dynamic nature, namely, the dependence on the Chern-Simons parameter and the invariant parameter of synchrotron radiation.

The goal of the fifth part of the work is the study of hydrodynamic problems. First, we investigate the problem of the flow of a heat-conducting viscous incompressible fluid in the field of gravity in a two-dimensional channel with heated walls with small periodic irregularities on them at large values of the Reynolds number. Then we investigate the problem of the flow of a viscous compressible gas (or liquid) in an axially symmetric pipe with small periodic irregularities on its surface at large values of the Reynolds number. The system of equations of gas dynamics without pressure is also investigated. This system is mentioned in the literature in two aspects. In the first aspect, as a model describing the distribution of density in rarefied dust clouds, in particular, in the Arnold-Zeldovich-Shandarin model describing the distribution of mass in the Universe. Another aspect is a new class of solutions discovered for these equations, more singular than those previously known in the equations of gas (hydro) dynamics of shock waves. These new solutions (alpha-shock waves) are arranged in such a way that the classical integral identities defining shock waves cannot be used to determine them. The aim of the work is to construct and study the limits of piecewise constant approximations of solutions of the Cauchy problem.

The goal of the sixth part of the work is to study exponential random graphs, i.e., a family of models originally proposed for analyzing and modeling of social networks. The probability of a certain graph configuration exponentially depends on the number of certain subgraphs (motifs) in the network. Such a model essentially describes a canonical ensemble of random networks with a given number of vertices. It was planned to study critical phenomena in network structures, including a wide range of phenomena (forms of behavior): structural variations in networks, appearance of a critical state, i.e., a scale-free network architecture, a variety of percolation phenomena (for example, origination of an epidemiological threshold), critical points in various optimization problems, and many others. Many of these critical phenomena are closely related, are of a similar nature, and admit of a universal description.

#### Methodology

The experimental modeling of quantum data transfer protocols was carried out remotely on quantum processors IBMqx4 and IBMqx5, the modeling of the associated system of Bose condensates of light and dark excitons was carried out using Belyaev diagram technique and generalized to a two-component system of Bogolyubov transformations, the study of the spin Hall effect for polaritons was carried out using diagonalization of the Hamiltonian of the exciton-photon system in external fields and the solution of transport equations, the radiation capture and superradiation in a superconducting system were simulated by numerical solution of Lindblad equation, the calculation of electron and phonon properties of layered electride was carried out by density functional method using the method of attached plane waves. The analysis of the properties of edge magnetoplasmons on graphene was carried out using the Wiener-Hopf method used to solve the electromagnetic problem and the self-consistent Born approximation used to calculate the conductivity. The superfluidity of a two-layer electron-hole system was studied in the mean field approximation, and the generalized quantum virial theorem for Dirac materials was analyzed using the apparatus of generalized currents and the method of scale transformations.

In the problem of quantum resonance-integrable systems with small symmetry violations, general methods of algebraic averaging of operators were used to obtain effective quantum Hamiltonians that describe the symmetry violations, as well as the WKB methods for differential and difference equations for semiclassical analysis of the spectrum of the obtained operators.

To construct formal asymptotic solutions to many large-scale problems of hydrodynamics with periodic perturbations, the previously developed approach based on a combination of the boundary layer decomposition method and the averaging method was used. Numerical simulation of flows in the wall region is based on the finite-difference approach. In the study of gas dynamics systems without pressure, the method of weak asymptotics was used to describe the interaction of stable and unstable elementary shock waves, and methods of functional analysis (properties of functions of bounded variation) were used to study the limit solutions.

Methods of contemporary statistical physics were used to analyze and model the complex networks. For computer modeling, we used Monte Carlo probabilistic methods, various algorithms on graphs, algorithms for finding communities in graphs (Newman methods, spectral analysis), algorithms for storing and working with big data, various algorithms for visualization of networks. A standard set of statistical physics methods was used for theoretical estimates, including the method of generating function, the mean field approximation, cluster analysis, and the theory of random matrices.

#### Empirical base of research

The results of research on the subject of the project published in the world scientific literature were used as the empirical base. IBM x4 and IBMqx5 quantum processors were remotely used for the experimental verification of the obtained results.

#### Results of research

In the first part of the work, the IBM quantum computers are used to illustrate the prospects of modeling quantum data transfer protocols to identify restrictions on the operation of quantum computers. The possibility of studying the Bose condensate of dark excitons indirectly via the interaction with the Bose condensate of light excitons is demonstrated. The spin Hall effect for excitons in two-dimensional dichalcogenides of transition metals in the field of two opposite laser beams is predicted. The possibility of realization of the effects of radiation capture and superradiation in a system of superconducting qubits connected with the resonator is predicted. The electron and phonon properties of the layered electride are investigated, and the strong anisotropy of its resistance is predicted. The theory of magnetoplasmons on graphene edges is constructed, which explains the suppression of their propagation velocity in the presence of dissipation and shielding. The possibility of controlling the superfluidity in a system of spatially separated electrons and holes using an external periodic potential is predicted. The generalized virial theorem and the Gelman-Feynman theorem for Dirac materials are obtained, and it is shown that, in such materials, the thermodynamic and kinetic pressures of the electron gas are different due to the anomalous contribution.

In the second part of the work, the spectrum of quantum resonance Hamiltonians with small symmetry violations is considered by an example of two classes of two-dimensional magnetic Schroedinger operators. For a particle in a strong magnetic field and in an electric field periodic in one direction and growing in the other direction, the semiclassical asymptotics of spectral zones arising in splitting of Landau levels is investigated. Asymptotics of exponentially small zones and gaps in the spectrum of the operator are obtained. Asymptotics of dispersion relations for low and high Landau levels are constructed. For a two-dimensional quantum Hamiltonian of a charge in a resonance electromagnetic trap, an effective quantum operator is obtained which asymptotically describes the anharmonic part of the Hamiltonian. A representation of the action-angle quantum coordinates in which the effective operator becomes a second-order difference operator is constructed. The semiclassical asymptotics of the spectrum and stationary states of this operator are obtained taking into account the description of exponentially small tunneling effects.

In the third part of the work, the symmetry algebra for a quantum three-frequency hyperbolic resonance oscillator is investigated, and it is shown that it can be given by a finite set of generators with polynomial commutation relations. Irreducible representations of this algebra and the corresponding coherent states are constructed.

In the fourth part of the work, an analytic expression for the anomalous magnetic moment of an electron was obtained and quantitative and qualitative results on the role of the magnetic field and the Chern-Simons parameter in studying the interaction energy of the anomalous magnetic moment of an electron with an external magnetic field in two-dimensional quantum electrodynamics were first obtained by analyzing this expression.

In the fifth part of the work, a system of equations describing the convective flow of a weakly compressible fluid in the wall region in a channel with periodic irregularities on the heated walls is obtained. A formal asymptotic solution for the flow of a compressible gas (or liquid) in an axially symmetric pipe with periodic irregularities on the wall is constructed. Numerical simulation of wall flows is carried out, the dependence of the flow character on the density of the main flow is investigated. The time-global approximation of solutions of the system of equations of gas dynamics without pressure in the form of piecewise constant functions with point masses at jump points is constructed. The limits of such approximations are investigated. The theorem of existence and uniqueness of the solution of the system of equations of gas dynamics with piecewise Lipschitz continuous initial conditions is proved. It is shown that the solutions of the Cauchy problem for the system under consideration do not have a semigroup property.

In the sixth part of the paper, a strong effect of finite size in exponential models is discovered. Numerical simulation allowed one to establish the existence of a critical size of the system depending on the density of a random graph. For systems whose size exceeds the critical value, a crossover from a random graph configuration to a previously unobserved clustered configuration is observed. This configuration is characterized by the formation of hubs, i.e., vertices with the highest possible degree and “steam”, i.e., weak interactions of the remaining vertices with each other. It is assumed that the formation of such a structure is due to the entropy trap of hubs. A model of exponential random graphs with various topological constraints and four-vertex interaction was constructed. Spectral properties of clustered states in exponential random graphs are investigated, in particular, the distance statistics between neighboring eigenvalues are constructed. The study of such statistics provides identification of excitation localization for the dynamics of one particle in space.

#### Level of implementation, recommendations on implementation or outcomes of the implementation of the results

The developed method of modeling the quantum data transfer protocols can be used for deep testing of the capabilities of contemporary and future quantum computers. The data of analysis of electronic and phonon properties of layered electride can be used in the development of electronic devices of new generation. The constructed theory of edge magnetoplasmons on graphene can be used for the development of plasmon chains. The predicted possibility of controlling the electron-hole superfluidity by the external periodic potential can be used to increase the critical temperature of transition of such a system to the superfluid state. The results of the study of resonant quantum systems can be applied in the modeling and development of quantum and nano-electronic devices. The results of the study of fluid flow in channels and pipes with heated rough walls can be used to study various processes of hydro- and gas dynamics and biomechanics.