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 Mathematical Methods in Theoretical Physics

- We explored the consequences of the new
*μ*-deformed generalization of the Heisenberg algebra for the coordinate and momentum operators obtained, within the Bose-condensate model of dark matter, from the condition of consistency of two*μ*-deformed analogues of the Lane-Emden equation with same solution, that gives an improved description of the density profile. Generalized uncertainty relation for the*μ*-deformed algebra, which at*μ*=0 reduces to well-known one, is derived and shown to depend on the states. As result, it leads to a quartet of maximal and minimal lengths and momenta. Their specific values, as well as temperature fluctuations, are obtained for the dark matter halo of a number of dwarf galaxies.*A. Gavrilik, A. Nazarenko* - Within the model of ultralight axionlike dark matter (DM) with periodic self-interaction, the existence of rarefied and dense phases of the Bose-Einstein condensate is revealed when describing the core of the DM halo of dwarf galaxies. This stems from two independent solutions of the Gross-Pitaevskii equation with the same model parameters. Based on the theory of Feshbach resonance for quantum-mechanical scattering of a pair with two channels (closed and open), we reveal a long-lived two-particle composite - a dimer - with a lifetime of millions of years, which is able to play a role in the formation of TM structures. Taking into account the additional (open) channel explains the empirically distinct values of the scattering length in the DM of different galaxies, but requires additional study of the internal structure of DM particles.
*A. Nazarenko, A. Gavrilik* - The dependence of the three-band structure consisting of a flat and two dispersion (upper and lower) bands, for the one-dimensional pseudospin-one Hamiltonian with a three-component potential, on the configuration of the components, each of these is constant, has been described. In the case, as the components of this potential have a rectangular shape, it is proven that the discrete spectrum of bound states crucially depends on the configuration of the rectangular component. This spectrum is shown to be one of the four characteristic types, which differ cardinally. For each type, the point interactions, which are given by corresponding connection matrices, have been constructed.
*V.P. Gusynin, A.V. Zolotaryuk, Y. Zolotaryuk* - Exact analytic expressions for the asymptotics of correlation function in the model of anyons on large distances and times in the spacelike and in the timelike regions were obtained. It turns out that in the timelike region the spectrum curvature plays an essential role and changes the asymptotic behaviour from pure exponential decay to exponential decay modified by a power prefactor.
*O.Gamayun, N. Iorgov, Yu. Zhuravlev* - For the elliptic Gaudin-type model in an external magnetic field associated with non-skew-symmetric elliptic r-matrix we have constructed a modified algebraic Bethe ansatz. The spectra of the relevant Gaudin-type Hamiltonians are found in terms of solutions of the modified Bethe equations. The applications of the obtained result to the diagonalization of the Hamiltonians of anisotropic quantum Euler top, quantum Zhukovsky-Volterrra top, quantum Steklov and Rubanovsky tops and of the generalized BCS-Richardson fermion Hamiltonian are given.
*T. Skrypnyk*

- Within the dark matter model based on the Gross-Pitaevski equation for a Bose-Einstein condensate with three-particle interaction, it is shown that at zero temperature the two- and three-particle correlations of the local density (averages of density products) indicate a first-order phase transition and reduce to the product of single-particle averages, as in an ideal gas, with increasing pressure. The time instability of two- and three-particle complexes is qualitatively described by quantum random walks between bound and unbound states. It is shown that the equations of chemical kinetics also admit the existence of complexes formed by dark matter particles with a long scattering length.
*A.M. Gavrilik, A.V. Nazarenko* - Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of multiple delta-function centers. One of these uses the Green's function method. The second approach is based on multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. Within both these approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.
*A.V. Zolotaryuk, V.P. Gusynin, O.O. Sobol, and Y. Zolotaryuk* - Effective form factors for one-dimensional lattice fermions with arbitrary phase shifts were introduced. Tau functions defined as series ізf these form factors were studied. On the one hand, the exact summation was performed and tau functions as Fredholm determinants in the thermodynamic limit were presented. On the other hand, simple expressions of form factors allow to present the corresponding series as integrals of elementary functions. Using this approach the asymptotics of static correlation functions of the XY quantum chain at finite temperature were rederived.
*O. Gamayun, N. Iorgov, Yu. Zhuravlov* - We have obtained a new integrable fermion hamiltonian of the BCS-Richardson type which is an anisotropic deformation of the famous fermion hamiltonian of Richardson, and have found its spectra with the help of the modified Bethe ansatz.
*T.V. Skrypnyk*

- Bose-condensate dark matter model is extended by taking into account both two- and three-particle self-interactions. The existence of nontrivial two-phase structure and possibility of first order phase transition in the central part of the dark matter halo of dwarf galaxies is confirmed. For a particular dwarf galaxy, the rotation curves and the entropy of bipartite entanglement in the halo of its dark matter are obtained.
*A.M. Gavrilik, A.V. Nazarenko* - Conditions under which a quantum particle can be described using classical quantities are studied. The wavefunction of a quantum particle submitted to a potential field for which all quantum effects vanish, even if Planck's constant is non negligible, is investigated. This problem is equivalent to the problem of the motion of a particle in a refringent medium. In these media, quantum particles have classical momenta, while their wave properties are described by the wave-optics equation. In the 1D case, the particle cannot be found in the region near the origin, since the index of refraction tends to infinity there. For the 3D case with central symmetry, the wave properties are determined by a function, that has a resonance of width of the order of the de Broglie wavelength.
*V.E. Kuzmichev, V.V. Kuzmichev* - The method of the algebraic Bethe ansatz was developed for the case of certain non-skew-symmetric gl(n)⊗gl(n)-valued classical r-matrices with spectral parameters. The Gaudin and BCS-type models associated with these r-matrices have been constructed. The spectrum and the eigen-vectors of the corresponding quantum Hamiltonians have been found in terms of solutions of the Bethe-type equations
*T.V. Skrypnyk* - The problem of variables separation is explored for the classical integrable hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Maillet brackets with spectral-parameter dependent a-b-c-d tensors. In terms of the corresponding a-b-c-d tensors components a sufficient condition that guarantees, that the separating functions of Sklyanin, Scott and Gekhtman produce canonical coordinates, is formulated. In the important subclass of classical a-b-c-d algebras, namely the case of classical reflection equation algebras, the analogous sufficient condition is formulated using the corresponding r-s-matrices, what guarantees the canonicity of the constructed coordinates. For the case of gl(n)xgl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition, we find a class of the Lax matrices for which the obtained set of canonical coordinates is complete.
*T.V. Skrypnyk* - 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 problem of variables separation is explored for the classical integrable hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Maillet brackets with spectral-parameter dependent a-b-c-d tensors. In terms of the corresponding a-b-c-d tensors components a sufficient condition that guarantees, that the separating functions of Sklyanin, Scott and Gekhtman produce canonical coordinates, is formulated. In the important subclass of classical a-b-c-d algebras, namely the case of classical reflection equation algebras, the analogous sufficient condition is formulated using the corresponding r-s-matrices, what guarantees the canonicity of the constructed coordinates. For the case of gl(n)xgl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition, we find a class of the Lax matrices for which the obtained set of canonical coordinates is complete.
*T.V. Skrypnyk* - We explore the consequences of including the repulsive three-particle interaction in the model of Bose-condensate dark matter model or fuzzy dark matter. Based on properly modified Gross-Pitaevskii equation, the model is aimed to describe the distribution of dark matter particles in the highly dense regions, such as the galaxy core and/or the overlapping halos of colliding galaxies. From numerical solutions for wave function and density profile, main thermodynamical characteristics are calculated. We reveal the existence of two distinct phases of dark matter, in the core of halo, separated by instability region between two special values of compression acting in the model. Some implications of the existence of two phases and the related first-order phase transition are discussed.
*A.M. Gavrilik, A.V. Nazarenko, M.V. Khelashvili* - 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*

- Model of dark matter, which was constructed on the basis of μ-deformed thermodynamics, has been developed to describe halo density profiles from the dark matter around dwarf galaxy. Herein, μ-deformed analogs of the Lane-Emden equation were constructed, and their solutions have been found. As result, the use of μ-deformation leads to improved agreement with observational data as compared to the well-known models (T. Harko and Navarro-Frenk-White ones). Possible physical meaning of the parameter μ is briefly discussed.
*A.M. Gavrilik, I.I. Kachurik, M.V. Khelashvili* - For the model of the slowly rotating Bose-Einstein condensed dark matter, under the equilibrium condition the partition function and macroscopic characteristics are computed and compared with the results following from the Gross-Pitaevskii equation solution. Using the found Green functions of operators of spatial evolution, the boundary conditions of the problem and the conditions of applicability of the Thomas-Fermi approximation are formulated in a unified way. The deformation of the commutation relations for the macroscopic wave function and the corresponding modification of the Gross-Pitaevskii equation yields a spatial distribution of dark matter, similar to the model with rotation, which describes well the observed data. Such an assumption about the properties of dark matter leads to the spatial inhomogeneity of entropy and the dependence of the distribution on temperature (below the Bose-condensation temperature).
*A.V. Nazarenko* - We consider the grand canonical ensemble of the static and extremal black holes, when the equivalence of the electric charge and mass of individual black hole is postulated. Assuming uniform distribution of black holes in space, we are finding the effective mass of test particle and mean time dilation at the admissible points of space, taking into account the gravitational action of surrounding black holes. Having specified the statistics that governs extremal black holes, we study its effect on those quantities. Here, the role of statistics is to assign a statistical weight to the configurations of certain fixed number of black holes. We borrow these weights from the Bose-Einstein, Fermi-Dirac, classical and infinite statis-tics. Using mean field approximation, the aforementioned characteristics are calculated and visualized, what permits us to draw the conclusions on visible effect of each statistics.
*A.M. Gavrilik, A.V. Nazarenko* - In view of the problem of separation of variables for integrable hamiltonian systems with
*gl(n)*-valued Lax matrices, satisfying quadratic Poisson brackets with skew-symmetric classical*r*-matrices, we have found, in terms of the corresponding r-matrices, the sufficient conditions which guarantee that the "magic recipe of Sklyanin" indeed produces canonical coordinates. An example of skew-symmetric trigonometric*r*-matrix, that satisfies the obtained conditions, is considered and a class of the Lax operators is found, for which the obtained set of canonical coordinates is complete.*T.V. Skrypnyk* - 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*

- Within the doubly special relativity theory which bounds both the velocities of particles (by the speed of light) and their energies (by the Planck scale) a new rule of so-called κ-addition for particle energies is proposed. A κ-dependent Hamiltonian of the one-mode multi-photon (sub)system is introduced and employed, with different modes being regarded as independent, to consider the thermodynamics of the black-body radiation and to obtain main thermodynamic quantities using analytical tools of the mean-field approximation and numerical evaluations based on exact formulas. The entropy of a one-mode subsystem turns out to be finite (bounded), and likewise the total energy density of radiation. Most unusual result is the existence of threshold temperature above which radiation is present.
*A.M. Gavrilik, A.V. Nazarenko* - A phenomenon of splitting (furcation) resonant tunneling under electron transmission through a plain layer in the form of a single delta-like barrier with at least one adjacent well located at non-zero sufficiently small distance has been observed. It is shown that in the zero-thickness limit of the heterostructure the detachment of a countable resonance curves from the background level occurs for certain critical value of the squeezing parameter.
*A.V. Zolotaryuk* - The Fredholm determinant and series representation of the tau-function of the Fuji-Suzuki-Tsuda system and its multivariate extension were derived. It is a generalization to higher rank (N=2) of the results obtained for Painlevé VI and the Garnier system. With its use a direct isomonodromic proof of the AGT-W relation for c=N−1 was obtained.
*N. Iorgov*

- Bose-condensate models of dark matter are rather popular, however possess some difficulties. We consider the μ-Bose gas model which is a deformation of Bose gas model (BGM) with rational type of nonlinearity, and at μ=0 reduces to the BGM. Within thermodynamic geometry of μ-BGM, the scalar curvature is calculated and its singular behavior is found. It confirms a presence of condensate in the system of μ-bosons. Using this fact and other important properties of μ-BGM which give some advantages relative to BGM, the application of the μ-deformed system for effective modeling of dark matter surrounding dwarf galaxies is proposed.
*A. Gavrilik, I. Kachurik, M. Khelashvili, A. Nazarenko* - For the one-dimensional Schrödinger equation, which describes the electron tunneling through a two- or three-layer heterostructure, in a single-point approximation, the effect of splitting resonant-tunneling levels into the countable sets, having the form of curves or surfaces that depend on two or three layers of the structure, has been observed and described in detail.
*A.V. Zolotaryuk* - An explicit solution of the rank N Fuji-Suzuki-Tsuda system with n singular points in terms of semidegenerate n-point conformal blocks of WN-algebra with central charge c=N-1 is constructed. This construction uses different properties of conformal blocks of WN-algebras. Although these properties are well known, some of them were proved for N>3 for the first time: (1) for arbitrary central charge a reduction of three-point conformal blocks to a minimal set of conformal blocks is given; (2) for c=N-1 for the degenerate field on the second level the restrictions on the fields to have non-zero three-point conformal blocks were found; (3) formula for the four-point conformal blocks with one semi-degenerate and one degenerate fields in terms of generalized hypergeometric functions is proved.
*P. Gavrylenko, N. Iorgov*

- Ideal D-dimensional Bose gas is considered in a trap (external potential) in more general case than in previous works, namely, for isotropic potential
*U(r) = Ar*, where^{n}+Br^{-m}*A, B*> 0. The momentum distribution function is obtained, as well as the total number of particles, critical temperature, condensate fraction and heat capacity. Both high and low temperature regimes are considered. Conditions for the existence of Bose-Einstein condensation in the form of certain relations on*n*,*m*and*D*are being obtained.*A.M. Gavrilik, I.I. Kachurik, Yu.A. Mishchenko* - The whole class of one-point interactions realized in the limit when the thickness of two- and three-layer planar heteostructures as well as the distance between them are infinitesimal. It is established that depending on the way of realization of this limit, in the tunneling of electrons, the phenomenon of furcation of isolated resonances into multi-valued sets takes place.
*A.V. Zolotaryuk* - We study symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are considered to be fixed different. The probability to find a walker at the given position and time is derived analytically. The probability distribution function is found: it has non-Gaussian form because of the properties of adsorption in the bulk of zones and reflection at the separation points. Time dependence of the mean squared displacement of a walker is also studied, and the anomalous behavior is revealed as compared with ordinary RW. Analytical calculation is in agreement with the results of numerical modeling based on first principles.
*A.V. Nazarenko*