List of Departments
Topmost Scientific Results

- Department of Astrophysics and Elementary Particles
- Department of High-Density Energy Physics
- Department of Theory of Quantum Processes in Nanosystems
- Department of Mathematical Methods in Theoretical Physics
- Department of Theory of Nonlinear Processes in Condensed Matter
- Department of Synergetics
- Department of Theory of Nuclei and Quantum Field Theory
- Department of Computer Maintenance

Department of Theory of Nonlinear Processes in Condensed Matter

- It is shown that introduced earlier new invariant of the Dirac equation with the Coulomb potential supplements the kown Dirac and Johnson-Lippmann invariants and they all form algebra. This allows to solve the equation using the algebraic method. It is found that different solutions of the equation are different presentations of the algebra of the spinor invariants. Within the group theory and using this algebra, relations between different presentations are found and all properties of the exact general solution are obtained in such algebraic approach. The hidden SU(2) symmetry (invariance with respect to the rotation in the Hilbert space of all eigen states) of the Dirac equation is demonstrated.
*L. Brizhik, A.A. Eremko, V.M Loktev* - An integrable nonlinear dynamical system of gauge-coupled intrasite excitations and lattice vibrations on an infinite one-dimensional regular lattice has been studied. Depending on the relationships between the two physically distinct spatial scaling parameters the system can manifest itself in three qualitatively distinct dynamical regimes referred to as the monopole regime, dipole regime, and threshold regime. The phenomenon of dipole-monopole crossover for the spatial distribution of pseudoexcitons is shown to initiate the partial splitting between the pseudoexcitonic and vibrational subsystems in the threshold dynamical regime specified by the threshold value of the localization parameter. This partial splitting is manifested by the complete elimination of one pseudoexcitonic component accompanied by the actual conversion of another pseudoexcitonic component into a pseudoexcitonic chargeless half mode. The integrable nonlinear pseudoexciton-phonon system under study is expected to be applicable for modeling the nonlinear dynamical properties of properly designed PT-symmetric metamaterials.
*O.O. Vakhnenko and V.O. Vakhnenko* - Within the framework of Green's function formalism, the pi-electron spectra of polyparaphenylene and polynaphthalene oligomers, which are the building blocks of armchair and chiral (2m,m) nanoribbons of arbitrary width, are found, analyzed and compared. In particular, it is shown that the spectrum of polynaphthalene (in contrast to polyparaphenylene) includes localized states.
*L.I. Malysheva* - The influence of discreteness on the fluxon interaction with the spatial dipole impurity in the Josephson transmission line is studied. This impurity is used to model the qubit inductively coupled to the Josephson transmission line. Fluxon assisted readout process of the qubit state is based on measuring the passage time it takes to pass through the dipole impurity. The discreteness of the medium plays an important role in this process. It was found that the fluxon delay time on the qubit impurity increases significantly if the discreteness of the medium increases, which is favorable for the sensitivity of the readout process. It is shown that the difference between the fluxon threshold currents for the qubit in different states decreases with the increase of discreteness, which in turn worsens the sensitivity of the readout process.
*I.O. Starodub, Y. Zolotaryuk* - The spectrum of spin waves propagating along a skyrmion string was obtained for the case when a spin-polarized current flows along the string. It was shown that the presence of the current leads to instability of the string translational mode. For an infinitely long string in an ideal ferromagnet, the instability effect is thresholdless and takes place for an infinitely small current.
*V.P. Kravchuk* - It is proposed and verified computationally, within the density functional approach, the concept of pre-fragmentation bonding that governs the ionization fragmentation in some amino acids under the influence of ionizing radiation.
*E.S. Kryachko* - Within the framework of the molecular dynamics simulations, the conformational space of natural polyamine molecules, interacting with the DNA double helix was studied. In particular, the system of DNA with spermidine3+ molecules was analyzed. The results showed that conformations of the spermidine3+ molecule (over 2000) can be classified into seven conformational modes. DNA macromolecule strongly influence the relative population of conformational states of polyamines, in particular two of the seven conformational modes of the spermidine3+ molecule appear only in complex with DNA. Interacting with DNA, the spermidine3+ molecules can significantly bend, inducing local conformational changes in the double helix. The obtained results are important for understanding the role of natural polyamines in the mechanisms of DNA biological functioning.
*Perepelytsya S.M.*

- The new invariant of the Diraq equation with the Coulomb potential is found. It supplements Diraq and Johnson-Lippman invariants, so that they together form algebra. It is shown that various solutions of the equation are various representations of this algebra. In frame of the group theory using the algebra of the spinor invariants the relation between different representations is found. The general solution of the Diraq equation is found within the algebraic approach. It is shown that the principal quantum number is not a mere combination of other quantum numbers, as it follows from the direct solving of the equation, but is a fundamental characteristic of the hydrogen atom.
*L. Brizhik, A.A. Eremko, V.M Loktev* - It is known that in two-dimensional relativistic Dirac systems placed in orthogonal uniform magnetic and electric fields, the Landau levels collapse as the applied in-plane electric field reaches a critical value $\pm E_c$. We study this phenomenon for a distinct field configuration with in-plane constant radial electric field. The Dirac equation for this configuration does not allow analytical solutions in terms of known special functions. The results are obtained by using both the WKB approximation and the exact diagonalization and shooting methods. It is shown that the collapse occurs for positive values of the total angular momentum quantum number, the hole (electron)-like Landau levels collapse as the electric field eaches the value +(-) E_c/2. The investigation of the Landau level collapse in the case of gapped graphene shows a number of distinctive features in comparison with the gapless case.
*S.G. Sharapov* - A one-dimensional nonlinear dynamical system of intra-site excitations and lattice vibrations coupled via gauge-like mechanism is studied. The system admits the semi-discrete zero-curvature representation and therefore it proves to be integrable in the Lax sense. Relaying upon an appropriately developed Darboux-Bäcklund dressing technique the explicit four-component analytical solution to the system is found and analyzed in details. Due to mutual influence between the interacting subsystems the physically meaningful solution arises as the essentially nonlinear superposition of two principally distinct types of traveling waves. The interplay between the two typical spatial scales relevant to these traveling waves causes the criticality of system's dynamics manifested as the dipole-monopole transition in the spatial distribution of intra-site excitations. The under-critical and over-critical regimes of system's dynamics are comprehensively illustrated graphically.
*O.O. Vakhnenko* - Using the solution of the Schrödinger equation with the Hückel-type Hamiltonian for carbon nanotubes (2m,m), analytical expressions for wave functions were obtained within the framework of the tight-binding model, and the relationship between the transmission coefficient and the band structure of chiral and achiral carbon nanotubes was analyzed. The influence of chirality on the coefficient of electron transmission through a system with a step-like potential is determined and it is shown that (2m,m) nanotubes with an average chiral angle exhibit intermediate transport properties compared to achiral (arnchair and zigzag) nanotubes.
*L. Malysheva* - A new class of topologically non-trivial solutions of the Landau-Lifshitz equation in the form of screw dislocations was found and investigated. The screw dislocations are realized in a helical phase of the cubic chiral magnets. A type of dislocation was found, which is continuously deformed into a skyrmion string under the action of an external magnetic field.
*V.P. Kravchuk* - A model of double helix deformation was built to describe changes in the macromolecule structure. In addition to elastic components (bending, twisting, stretching), the presented model includes conformational rearrangements of the macromolecule, as well as describes the relationship between the external and internal degrees of freedom of the DNA structure. The obtained shapes, values and energy of the DNA deformation sites allow us to propose a probable mechanism of recognition of key DNA sequences due to their deformation, as well as to explain the high accuracy of genetic information reproduction processes.
*P.P. Kanevska, S.N. Volkov* - Using the density functional theory method in the presence of a physiological environment the structure and energy of formation of atomic molecular complexes are determined groups of nucleic bases of the DNA macromolecule with hydrogen peroxide molecules and water It is shown that hydrogen peroxide forms stable complexes with atomic grains of DNA, blocking the centers of specific recognition of the macromolecule.
*D.V. Pyatnytskyi*

- A new spin relevant invariant of the Dirac equation with the Coulomb potential is constructed. It does not commute with the known Dirac and Johnson-Lippman invariants and defines the spin degree of freedom. The general solution of the Dirac equation is found. It is described by the full set of quantum numbers which characterize hydrogen-like energy states. Spatial distribution of the charge density and spin orientation meanvalue are calculated and shown to depend significantly on the spin invariant. This can be manifested experimentally.
*Acad. of the NAS of Ukraine V.M. Loktev, L. Brizhik, O. Eremko* - The nonlinear dynamics of coupled PT-symmetric excitations and Toda-like vibrations on a onedimensional lattice are studied analytically and elucidated graphically. The nonlinear exciton-phonon system as the whole is shown to be integrable in the Lax sense inasmuch as it admits the zero-curvature representation supported by the auxiliary linear problem of third order. Inspired by this fact, we have developed in detail the Darboux-Bäcklund integration technique appropriate to generate a higher-rank crop solution by dressing a lower-rank (supposedly known) seed solution. In the framework of this approach, we have found rather non-trivial four-component analytical solution exhibiting the crossover between the monopole and dipole regimes in the spatial distribution of intra-site excitations. This effect is inseparable from the pronounced mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two essentially distinct types of travelling waves. We have established the criterion of monopole-dipole transition based upon the interplay between the localization parameter of Toda mode and the intersubsystem coupling parameter.
*О.О. Vakhnenko* - Using solution of the Schroedinger equation with Hukkel type Hamiltonian for carbon nanotubes (2m,m) the analytical expressions for the wavefunctions are obtained in the model of strong coupling. The relation between the transmission coefficient and band structure of chiral and non-chiral nanotubes is analyzed. It is found the impact of non-chirality on the transmission coefficient for electrons across the system with the step-like potential. Range of parameter values is given, at which experimental observation of chirality impact on transport characteristics of carbon nanotubes is obtained.
*L.I. Malysheva* - Spin waves of skirmion lattices in many-layered ferromagnetic heterostructures are studied theoretically and experimentally. It is shown that strong dipole-dipole interaction leads to significant change of the positions of magnon modes which have non-zero magnetic moment. From the comparison of experimental and theoretical evolutions of spectra in the magnetic field Dzyaloshinsky-Morria coupling constant is calculated for the heterostructure Ir/Fr/Co/Pt.
*V.P. Kravchuk* - The structure and energy formation of molecular complexes of DNA atomic groups of sugar-phosphate backbone with molecules of hydrogen peroxide and water are studied. Using the method of molecular mechanics, approaches of quantum chemistry and the presence of a physiological environment, the interaction energies are calculated and the spatial configurations of molecular complexes of peroxide with phosphate of DNA backbone are optimized. Calculations show that under physiological conditions, hydrogen peroxide can stay near phosphate groups of DNA for a long time, blocking the centers of genetic activity of the macromolecule.
*D.V. Piatnytskyi, O.O. Zdorevskyi, S.N. Volkov* - The formation of a double helix of DNA in aqueous solution with hydrogen peroxide molecules and Na + counterions is modeled. Molecular mechanics shows that hydrogen peroxide molecules bind to the oxygen atoms of the phosphate groups of the double helix and thus replace the water molecules of the hydrated shell of the macromolecule. Three layers of hydrogen peroxide ordering around the phosphate groups of the DNA helix with centers of density around 2.6 and 3.3 angstroms, as well as a wide band with a center around 5 angstroms were determined. Thus, hydrogen peroxide molecules that interact with phosphates can inhibit the formation of hydrogen bonds necessary for the biological functioning of DNA.
*S.N. Volkov, S.M. Perepelytsya* - Equations for the scattering data and discrete spectrum were obtained for the one-dimensional Schroedinger equation with the piecewise-constant potential. The asymptotical three-scale method of squeezing the system to zero width has been developed. It has been shown that the boundary conditions in the singular point make sense only when the mutual divergence cancellation in the connection matrix is possible. Two types of the transcendental equations for the resonance sets have been derived.
*O.O. Zolotaryuk, Y.O. Zolotaryuk* - The dispersion law for plasmons in the quasi-one-dimensional array of Josephson junctions, that consists of the finite number of rows N and the infinite numbers of columns, is derived. If external bias is absent then the spectrum has a N-fold degenerate flat band. The degeneracy is lifted when the external bias is applied and, as a result, only one flat branch remains.
*D.V. Bukatova, Y.O. Zolotaryuk*

- The description of the collective magnetorheological effect induced by magnetic field in elastomers with uniaxial ferromagnetic particles is proposed. The condition of consistency is used between magnetic and mechanic momenta of forces exerted on these particles in elastomer at their magnetization. The study shows that even in the case of their small concentration, the value of magnetically-induced shear can be anomalously large, reaching up to tens of percent. The deformation of magneto-active elastomer can evolve critically as a second-order phase transition if magnetic field is aligned along the easy axis of particles.
*Acad. of the NAS of Ukraine V.M. Loktev* - A novel family of disordered systems is proposed. This family belongs to the class of systems containing random substitutional non-Hermitian impurities. The consideration is limited by a rather simple case when the presence of substitutional point defects results in the model Hamiltonian featuring a diagonal disorder. In contrast to known models of non-Hermitian impurities, the nonzero density of states for each isolated from the host impurity is restricted to a continuous band of finite width. A method to construct corresponding impurity Hamiltonians is provided.
*Yu.V. Skrypnyk, Acad. of the NAS of Ukraine V.M. Loktev* - Using the methods of density functional and molecular mechanics, it has been shown that hydrogen peroxide molecules form stable complexes with the phosphate group of the DNA macromolecule. Under physiological conditions, such complexes are more energetically favourable than similar complexes with a water molecule. The energy advantage for complexes with hydrogen peroxide is achieved, in particular, due to the presence in the molecule of a torsional degree of freedom, which makes it more flexible compared to the water molecule. These results show that in living tissues, hydrogen peroxide can remain near the phosphate groups of DNA for a long time, blocking the genetic activity of the macromolecule.
*S.N. Volkov, D.V. Piatnytskyi, O.O. Zdorevskyi* - Two families of point interactions with bound state energy are realized from a heterostructure composed of two parallel homogeneous layers in the limit as their width and the distance between them tend to zero simultaneously. Under certain conditions described by transcendental equations, a resonant-tunneling transmission of electrons through this limit structure is shown to occur. A particular example of the singular potential having the form of the derivative of the Dirac delta-function is generalized to a whole family of point interactions, for which a single bound state does exist, contrary to the widespread opinion on the non-existence of bound states in δ-like systems.
*A. Zolotaryuk, Ya. Zolotaryuk* - The nonlinear model of three dynamical subsystems, coupled both in their kinetic and potential parts, has been suggested. Due to the quasi-one-dimensional spatial structure of its underlying lattice the model grasps several degrees of freedom capable to imitate the dynamical behavior of long macromolecules both natural and synthesized origins. The model admits a clear Hamiltonian formulation with the standard form of fundamental Poisson brackets and it demonstrates the complex-conjugate symmetry between two subsystems of a Toda type. The integrability of system equations is supported by their zero-curvature representation based upon the auxiliary linear problem with the relevant spectral operator of third order. In view of these facts, we have developed an appropriate two-fold Darboux-Bäcklund dressing technique capable to generate the nontrivial crop solution embracing all three coupled subsystems. The explicit crop solution to the nonlinear model is found to be of a pronounced pulson character.
*O.O. Vakhnenko* - The theory of curvilinear one-dimensional antiferromagnets has been developed. It has been established that the achiral antiferromagnetic spin chain behaves like a biaxial chiral magnet with effective anisotropy and Dzialoshinsky-Moria interaction. The geometry-induced Dyaloshinsky-Moria interaction leads to hybridization of magnons in the chain.
*K.V. Yershov, V.P. Kravchuk* - We study the manifestation of the Nernst effect in the Corbino disk subjected to the normal external magnetic field and to the radial temperature gradient. The Corbino geometry offers a precious opportunity for the direct measurement of the magnetization currents that are masked by the kinetic contributions to the Nernst current in the conventional geometry. The magnetization currents, also referred to as the edge currents, do not depend on the conductivity of the sample which is why they can be conveniently described within the thermodynamic approach. They can be related to the Landau thermodynamic potential for an infinite system. We demonstrate that the observable manifestation of this, purely thermodynamic, Nernst effect occurs as strong oscillations of the magnetic field measured in the center of the disk as a function of the external field. The oscillations depend on the temperature difference at the edges of the disk. Dirac fermions and 2D electrons with a parabolic spectrum are characterized by oscillations of different phase and frequency. We predict qualitatively different power dependencies of the magnitude of the Nernst signal on the chemical potential for normal and Dirac carriers.
*S.G. Sharapov*

- Based on the Dirac quantum theory of the spinor field, the generelized operator of the spin-orbit interaction is found. It is shown that this operator contains not only the known Thomas-Frenkel correction, but also a new term, so that the consistent description of the influence of the spin-orbit interaction on electrons in the external potential in frame of the non-relativistic Schroedinger equation is possible only with the account of this generelized operator.
*L.S. Brizhik. O.O. Eremko, Acad. of the NAS of Ukraine V.M. Loktev* - An explanation is proposed for the recently observed a giant increase of the Drude peak width under applied uniaxial strain in optical spectra of monolayer graphene. The underlying mechanism of this increase can be based on resonant scattering of carriers from inevitably present impurities such as adsorbed atoms that can be described by the Fano-Anderson model. It is demonstrated that the often neglected scalar deformation potential plays the essential role in this process. The conditions necessary for the maximum effect of the giant Drude peak broadening are determined.
*V.O. Shubnyi, Yu.V. Skrypnyk, S.G. Sharapov, Acad. of the NAS of Ukraine V.M. Loktev* - Two types of general nonlinear integrable systems on infinite quasi-one-dimensional regular lattices are proposed. The method is suggested which allows to unify the Toda-like vibration subsystem and the self-trapping-like exciton subsystem into a single integrable system, thereby substantially extending the range of realistic physical problems that can be rigorously modeled. Several lowest conserved densities associated with either of the relevant infinite hierarchies of local conservation laws are found explicitly in terms of prototype field functions.
*O.O. Vakhnenko* - The properties of embedded solitons (solitons that exist despite their resonance with the linear spectrum of the system and have only isolated values of velocity) in the discrete double sine-Gordon equation with next-neighbor and second-neighbor interactions are investigated. This model describes the DC biased array of the superconductor/ferromagnet/ superconductor junctions. Depending on the sign of these interactions they can be either destructive or favorable for the embedded soliton creation. The embedded soliton existence area depends on the width of the linear spectrum: narrowing of the spectrum widens the embedded soliton existence range and vice versa.
*Y. Zolotaryuk, I. Starodub* - An approach to define single-point interactions under the application of external fields has been developed. The essential feature is an asymptotic method based on the one-point approximation of multi-layered heterostructures that are subject to bias potentials. In the zero-thickness limit, the matrix connecting the two-sided boundary conditions of wave function at the point of singularity has been obtained. The dependence of the reflection and transmission coefficients on the bias potential has been studied. Several "one-point" models of two- and three-terminal devices in the semiconductor physics have been elaborated. The notion of a "point" transistor has been introduced, and the existence of extremely sharp peaks as an applied voltage tunes, forming a resonance set for this one-point interaction, is shown.
*A.V. Zolotaryuk, Y. Zolotaryuk* - The interaction of hydrogen peroxide and water molecules with the regions of specific and non-specific recognition of DNA molecules by proteins is analyzed. These interactions are important for explanation of beam therapy of cancer. On the basis of density functional theory, the interaction energies of water and hydrogen peroxide molecules with DNA nuclear bases are calculated and optimal geometries of stable complexes are determined. The positions are found where binding of hydrogen peroxide molecule exceeds the one for water molecule.
*S.N. Volkov, D.V. Piatnytskyi, O.O. Zdorevskyi* - The interaction of polyamines with the DNA macromolecule was studied using molecular dynamics simulations. The polyamines molecules were found to be localized mostly in the minor groove of the double helix in the region with the specific sequence of nucleotides of A-tract type. The results of the study agree with the existing experimental data and explain the preferential localization of polyamines molecules in the DNA sites with characteristic nucleotide sequence.
*S.M. Perepelytsya*

- It is shown that inhomogeneous curvature of magnetic wire induces motion of a domain wall in the direction of the curvature gradient increasing. An asymptotic domain wall velocity is found. The resuts are verified by micromagnetic simulations.
*Yu.B. Gaididei, V.P. Kravchuk, K.V. Yershov* - The two-stage mechanism of the appearance of threshold deformations in a DNA macromolecule under the action of external force is proposed. At the first stage the appearance of a bistable state and the formation of domains with different conformations in the macromolecule chain occur for some critical value of external force. On the second stage the cooperative process of the domain walls propagation happens in the macromolecule. Thus, the emergence of bistability causes the threshold effect of deformation. The calculated parameters of the threshold elongation of DNA double helix are consistent with the experimental data.
*S.N. Volkov* - The character of hydration of Na+, K+, Cs+ and Mg2+ counterions interacting with DNA double helix is studied using molecular dynamics simulations. The results show that structuring of the first and the second hydration shells of the ions is essential for the specific interactions with DNA. In particular, Na+ and Mg2+ interact with the DNA mostly via water molecules of the first hydration shell, while K+ and Cs+ may be dehydrated and penetrate deep into the minor groove of the double helix.
*S.M. Perepelytsya* - It is shown that with the accuracy of the second order relativistic corrections the Hamiltonian of particles and antiparticles in an external field can be reduced to the Hamiltonian of non-interacting electrons and positrons. Operator invariants are obtained which describe spin states of relativistic particles. The Hamiltonian of electrons in the non-relativistic approximation is shown to contain both known and new relativistic corrections.
*A.A. Eremko, L.S. Brizhik, Acad. of the NAS of Ukraine V.M. Loktev* - Discrete time nonlinear quantum walk dynamics are studied. In the linear limit the spectrum of this model can be reduced to the complete flat band shape, what means the absence of transport and presence of the compact localized states. Existence of the soliton-like exitations that can propagate with constant velocity has been demonstrated. Due to the flatness of the linear spectrum the soliton does not resonate with it. The absence of the linear waves also causes the super-exponential [~exp(-exp(x))] decay of the soliton amplitude.
*Ya. Zolotaryuk* - The new nonlinear integrable system of classical intra-molecular excitations on a ladder lattice with three structural elements in the unit cell is suggested. The Poisson structure and the Hamiltonian formulation of the system are found. In the framework of Darboux transformation for an auxiliary spectral problem the system's multicomponent soliton solution is obtained.
*O.O. Vakhnenko*

- A new mechanism of point mutation formation in biological systems that takes into account the fluctuation appearance of complementary pairs preopened by water in the DNA double helix has been proposed. The possible proton transitions in the preopened A-T pair have been analyzed within the density functional theory, and their energies have been numerically obtained. It is shown that under formation of an opened pair, irregular tautomeric forms of the nucleic bases can be stabilized and be the origin of point mutations. The estimated probability of the occurrence of these mutations in DNA falling in the interval of 10
^{-10}-10^{-11}explains the known experimental facts.*E.S. Kryachko, S.N. Volkov* - The mixed set of primary and intermediate fields for the integrable nonlinear Schrödinger system on a triangular ladder lattice is found out. Having been written in terms of these field variables the system is shown to split into two subsystems with qualitatively different properties of excitations. The crossover effect in the system dynamical behavior at the critical value of background parameter is established.
*О.О. Vakhnenko*

- It is known that the Shubnikov - de Haas oscillations can be observed in the Hall resistivity, although their amplitude is much weaker than the amplitude of the diagonal resistivity oscillations. Employing a model of two-dimensional massive Dirac fermions that exhibits anomalous Hall effect, we demonstrate the amplitude of the Shubnikov - de Haas oscillations of the anomalous Hall conductivity is the same as that of the diagonal conductivity. We argue that the oscillations of the anomalous Hall conductivity can be observed by studying the valley Hall effect in graphene superlattices and the spin Hall effect in the low-buckled Dirac materials.
*S.G. Sharapov* - It is shown that the curvature and torsion of a ferromagnetic nanowire significantly affect the spin-polarized current induced domain wall motion along the wire. We consider the simplest case of constant curvature and torsion, corresponding nanowires in the form of three-dimensional helix. It is shown that the curvature leads to Walker breakdown, i.e. the emergence of a critical density of spin-polarized current j
_{c}, which separates two modes of motion: translational motion when j<j_{c}and oscillating motion for j> j_{c}. In the point j = j_{c}a saddle-node bifurcation takes place. The nanowire torsion, in turn, leads to a shift of the material parameter of spin torque nonadiabaticity. The shift is proportional to the product of the helix chirality to the topological charge of the domain walls. This shift can be negative, that can lead to negative domain wall mobility. Basing on the obtained results we propose a practical method of experimental measurement of the nonadiabaticity parameter and the damping coefficient.*Yu.B. Gaididei, V.P. Kravchuk, K.V. Yershov* - The analytical expression for the spectrum of electrons in quasi-two-dimensional space (so called 2D electrons) is found within the fundamental Dirac equation. Such system describes layered heterostructures, interfaces, crystal surfaces, etc. It is shown that the exact solution contains several spin invariants, which means possibility of the existence of many spin states of quasi-2D electrons with different spin polarization and different effects of spin-orbit interaction on the energy spectrum. The exact solution is found for electrons confined by asymmetric potential well, which includes free parameters, whose value corresponds to certain invariants. In the general case the spin quantum number determines certain value of the linear combination of all invariants. These results allow to ascertain the conditions at which the given spin state is realized, which opens new perspectives to control the spin degree of freedom of charge carriers in spintronics.
*A.A. Eremko, V.M. Loktev, L.S. Brizhik* - We present the analytic analysis of the spectral problem for bounded graphene fragments and finite polyene chains. The solutions of the transcendental equation for the characteristic quantum numbers are thoroughly examined and their accurate approximations are proposed. The obtained results are used for analysis of the fundamental and technologically significant properties of graphene and its daughter structures (i.e., zigzag and armchair graphene nanoribbons), and finite polyene chains of various length.
*L.I. Malysheva* - The variativity of governing coupling parameters in the integrable nonlinear Schrödinger system on a triangular-lattice ribbon is shown to ensure the important qualitative rearrangements in the system dynamics. Thus the longitudinal coupling parameters regulated mainly by the background values of concomitant field variables are responsible for the bifurcation of primary integrable nonlinear system into the integrable nonlinear system of Ablowitz-Ladik type. As a consequence in a critical point the number of independent field variables is reduced by a half and the system Poisson structure turns out to be degenerate. On the other hand the transverse coupling parameters regulated basically by the choice of their a priori arbitrary dependencies on time are capable to incorporate the effect of external linear potential. As a consequence the primary integrable nonlinear system with appropriately adjusted parametrical driving becomes isomorphic to the system modelling the Bloch oscillations of charged nonlinear carriers in an electrically biased ribbon of triangular lattice.
*O.O. Vakhnenko* - Molecular force field (CHARM), which is commonly used in computer simulations to study the structure and dynamics of macromolecules in biological systems, is modified. A new type of potential function allows us to study the processes of chemical bonds breaks, the rupture of valence angles and dihedral interactions in molecules. Modified force field implemented in the software package MBN Explorer.
*S.N. Volkov* - The structure functions of the mirror nuclei
^{14}С and^{14}O are studied within a model "three alpha-particles plus two extra nucleons". The charge density distributions, form factors, pair correlation functions, and the momentum distributions of particles are found for these nuclei. Two spatial configurations are revealed in the ground state of^{14}С and^{14}O nuclei. The unknown experimental charge r.m.s. radius of^{14}O nucleus is predicted.*B.E. Grinyuk, D.V. Piatnytskyi*