A new order parameter is used to construct phase diagrams of Stockmayer models for multiparticle systems with anisotropic pair potential generated by dipole moments of individual particles. It is shown that there is a liquid-crystal phase of such a system, and the range of the system parameters where this phase can be observed is studied. The obtained results testify that the liquid crystals of this type have nontrivial (compared with other types of liquid crystals) topological and symmetry properties, in particular, they have three typical spatial periods.

It is shown that the frequencies of TE-waves in graphene lie in a narrow interval whose lower boundary very strongly depends on the value of the optical contrast. A principle of an optical volumetric supersensitive sensorics is proposed, which is convenient to detect and analyze volume matters attached to a surface, for example, various biological objects. The proposed graphene sensor has the same resolution as the optical sensors produced on the basis of the Fabry-Perot resonances but exceeds them in sensitivity by three orders.

The spaser produced on the basis of graphene ribbons and semiconductor nanostructures is analyzed theoretically. The dependence on parameters is calculated for the minimal inverse population of semiconductor quantum points which is required to implement the generation mode of the spaser. The proposed realization of the spaser on the basis of graphene nanoribbons has several advantages over the traditionally used spasers on metals, namely, a wide generation range from the teraherz to the infrared region, weak attenuation, a low generation threshold, and the possibility of the parameter retuning due to the electrostatic doping.

The characteristics of resonance gaps in the spaser generation line arising because of the interaction with the sample were calculated as functions of the number of atoms in the sample and their position with respect to the tip of the scanning needle. The optimal conditions for scanning the sample and studying its absorption spectrum were analyzed. A spectro-microscopy method based on the use of a spaser in the form of the scanning microscope needle with semiconductor quantum points distributed near it was proposed. The proposed methodology has both the advantages of the probe-reinforced optical microscopy and intracavity laser spectroscopy. This methodology permits studying supersmall (dozens of atoms) amounts of matter with nanometer spatial resolution and simultaneously obtaining some information about the composition of matter.

A model of the quantum Penning nanotrap with a perturbed magnetic field was analyzed mathematically in the regime of the principal resonance of normal frequencies. The effect of the secondary resonance was discovered and the secondary algebra or symmetries of the trap was presented (this algebra is of non-Lie type, i.e., is defined by nonlinear commutation relations). The irreducible representations of this algebra were constructed; the coherent states and the transformation of the initial three-dimensional trap Hamiltonian to a one-dimensional differential operator of the second order were described. These results can be used, for instance, to construct models of quantum computers.

The effective Hamiltonian describing the splitting of Landau levels and the quantum states of Larmor quasiparticles in curved nanofilms and in graphene in a strong magnetic field was constructed. The quantum coordinates of the «leading center», which takes the curvature and noncommutativeness of the polarization into account, were found. The numerically stable representations of the solutions to problems of quantum dynamics, which are based on the use of squeezed states of quantum optics, were justified.

The tunnel splitting of the spectrum of the Schroedinger operator in nonsymmetric double well potential was calculated. A criterion for the resonance tunneling, which reflects the relationship between the bilocalization of eigenfunctions and the exponential splitting of the energy levels, was obtained. The results are important for the models of qubits, Josephson junctions, and problems of spectroscopy of molecules.

A model of heat propagation in field emission spike (conical) nanocathodes, which also described the melting effects, was developed. The parameters significantly affecting the heat transport process via the Nottingham effect were discovered. The developed computer program can be used to design new technical devices like electron microscopes, FED screens, micro-CRT.

Computer simulation of dusty plasma was performed by using an extended system of equations in the near-electrode layer of gas discharge, which takes the dust particle charge fluctuations and the charge dependence on the distance from the electrode and to the other particles into account. It was shown that these effects significantly affect the system. An analysis of the simulation results was used to propose a new mechanism for increasing the mean kinetic energy of dust particles in the gas-discharge plasma and the energy transfer in the dusty plasma system.