Various electronic, mechanical and optical phenomena in nano- and low-dimensional structures based on graphene, borophene, metals, and semiconductor were studied. In particular, generalized virial theorem for a gas of massless Dirac electrons with Coulomb interaction was stated and proved. This theorem provides an exact relation between different characteristics of the electron gas and has deep connections with scale transformations. Pseudoelectric and pseudomagnetic fields which effectively act on massless Dirac electrons in a novel two-dimensional material borophene under its deformations are predicted and analyzed. Different realizations of hyperbolic metasurfaces are studied: an array of graphene nanoribbons, a two-dimensional array of uniaxial metallic nanoparticles and the black phosphorus. Considerable effects of polarization plane rotations on these surfaces in terahertz, infrared and visible ranges are predicted. A number of additional results having fundamental and practical importance are also obtained.

An incompressible viscous fluid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thin boundary layer (``lower deck'') and a classical Prandtl boundary layer (``upper deck''). It is shown that the existence and uniqueness of the solution of a Rayleigh-type equation, which describes which describes oscillations of the vertical velocity component in the classical boundary layer, reduces to proving that the discrete spectrum of a Schrödinger-type operator with potential in the form of a small well is empty. Also, the asymptotic solution with double-deck structure is constructed for a case of a compressible fluid, and the results of numerical solution of the thin boundary layer equations are shown. We proposed a method of constructing a weak asymptotic solution for the system of gas dynamics equation, which is based on the difference schemes designed in continuous time. The theorem of existence a weak asymptotic solution of Euler equations on torus is proved.

Analytical description of the cluster formation is random graphs of different topology is built. Random Erdos-Renyi graphs, random regular graphs, bimodal graphs, scale-free graphs are studied. We consider the canonical ensemble of Erdos-Renyi graphs with quenched vertex degree and with fugacity µ for each closed triple of bonds. We claim complete defragmentation of such topological graphs into the collection of almost full subgraphs (cliques) above some critical value µ_{с}. The number of cliques above µ_{с} behaves as 1/p, where p is the ER node formation probability. The phenomenon is analyzed in terms of the evolution of the spectral density, of the adjacency matrix for various µ. . Evolution of the spectral density, of the adjacency matrix with

increasing leads to the formation of a multi-zonal support. Eigenvalue tunnelling from the central zone to the side one means formation of a new clique in the defragmentation process. The adjacency matrix of the network ground state has a block-diagonal form, where the number of vertices in blocks fluctuate around the mean value Np. The spectral density of the whole network in this regime has triangular shape.

The model of the planar Penning trap for a single electron is studied. The axial symmetry is broken by deviation of the magnetic field from the trap axis by a small angle (the small parameter in this model). Geometry of the planar electrodes, as well the electric voltage on them and the strength of the magnetic field are consistent in order to obtain a combined resonance< in the prime and subprime Hamiltonians under the small parameters expansion. It is demonstrated that due to this resonance the classical microprap demonstrates quantum properties: well observed discrete spectrum and the opportunity for tunneling transfers. A mathematical technique is developed allowing to compute the tunnel spectral splitting which controls the time of the transfer.

We study the asymptotics of solutions of linear difference equations (recurrence relations) with slowly varying coefficients. It is known that the local asymptotic behavior of solutions can be obtained similarly to the WKB approximation for linear differential equations. In contrast to the continuous case, one of the major obstacles to the widespread use of discrete WKB method is the lack of a geometric interpretation of the obtained asymptotic formulas. We show that it is possible to build a simple geometric interpretation of discrete WKB method if one consider the difference equation as pseudo-differential with corresponding Weyl symbol (Hamiltonian). We obtain such a geometric interpretation for local asymptotics, turning points, the Bohr-Sommerfeld rule and other basic elements of the semiclassical approximation.