
Laboratory of Astrophysics and Cosmology

Laborarory of Theory of Integrable Systems

Laboratory of Stronglycorrelated Low Dimensional SystemsLaboratory of Biophysics of Macromolecules

Laboratory of Mathematical Modelling

Laborarory of Structure of Atomic Nuclei
Department of Mathematical Methods in Theoretical Physics
Deputy head of department – Prof., Dr. Oleksandr M. Gavrilik.
The department incorporates the Laboratory of Integrable Systems.
Head of the laboratory – Dr. Nikolai Z. Iorgov.
Research fields
 Symmetry theory: methods of the theory of groups, their representations, and physical applications
 Quantum groups, quantum algebras, quantum homogeneous spaces
 Group representations theory and special functions
 Theory of integrable systems
 Classical and quantum field theory
 Theory of elementary particles
Laborarory of Theory of Integrable Systems
 We explored the consequences of the new μdeformed generalization of the Heisenberg algebra for the coordinate and momentum operators obtained, within the Bosecondensate model of dark matter, from the condition of consistency of two μdeformed analogues of the LaneEmden equation with same solution, that gives an improved description of the density profile. Generalized uncertainty relation for the μdeformed algebra, which at μ=0 reduces to wellknown 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 selfinteraction, the existence of rarefied and dense phases of the BoseEinstein condensate is revealed when describing the core of the DM halo of dwarf galaxies. This stems from two independent solutions of the GrossPitaevskii equation with the same model parameters. Based on the theory of Feshbach resonance for quantummechanical scattering of a pair with two channels (closed and open), we reveal a longlived twoparticle 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 threeband structure consisting of a flat and two dispersion (upper and lower) bands, for the onedimensional pseudospinone Hamiltonian with a threecomponent 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 Gaudintype model in an external magnetic field associated with nonskewsymmetric elliptic rmatrix we have constructed a modified algebraic Bethe ansatz. The spectra of the relevant Gaudintype 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 ZhukovskyVolterrra top, quantum Steklov and Rubanovsky tops and of the generalized BCSRichardson fermion Hamiltonian are given.T. Skrypnyk
 Within the dark matter model based on the GrossPitaevski equation for a BoseEinstein condensate with threeparticle interaction, it is shown that at zero temperature the two and threeparticle correlations of the local density (averages of density products) indicate a firstorder phase transition and reduce to the product of singleparticle averages, as in an ideal gas, with increasing pressure. The time instability of two and threeparticle 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 onedimensional Dirac equation with the potential consisting of multiple deltafunction centers. One of these uses the Green's function method. The second approach is based on multiplying the onecenter 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 deltacenters and the distance between them. The principle of strength additivity is analyzed in the limits as the deltacenters 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 onedimensional 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 BCSRichardson 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
 Bosecondensate dark matter model is extended by taking into account both two and threeparticle selfinteractions. The existence of nontrivial twophase 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 waveoptics 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 nonskewsymmetric gl(n)⊗gl(n)valued classical rmatrices with spectral parameters. The Gaudin and BCStype models associated with these rmatrices have been constructed. The spectrum and the eigenvectors of the corresponding quantum Hamiltonians have been found in terms of solutions of the Bethetype equationsT.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 spectralparameter dependent abcd tensors. In terms of the corresponding abcd 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 abcd algebras, namely the case of classical reflection equation algebras, the analogous sufficient condition is formulated using the corresponding rsmatrices, what guarantees the canonicity of the constructed coordinates. For the case of gl(n)xgl(n)valued trigonometric abcd 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 onedimensional Schroedinger equation with the piecewiseconstant potential. The asymptotical threescale 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 spectralparameter dependent abcd tensors. In terms of the corresponding abcd 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 abcd algebras, namely the case of classical reflection equation algebras, the analogous sufficient condition is formulated using the corresponding rsmatrices, what guarantees the canonicity of the constructed coordinates. For the case of gl(n)xgl(n)valued trigonometric abcd 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 threeparticle interaction in the model of Bosecondensate dark matter model or fuzzy dark matter. Based on properly modified GrossPitaevskii 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 firstorder 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 resonanttunneling 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 deltafunction is generalized to a whole family of point interactions, for which a single bound state does exist, contrary to the widespread opinion on the nonexistence of bound states in δlike systems.A. Zolotaryuk, Ya. Zolotaryuk
 Model of dark matter, which was constructed on the basis of μdeformed thermodynamics, has been developed to describe halo density profiles from the dark matter around dwarf galaxy. Herein, μdeformed analogs of the LaneEmden equation were constructed, and their solutions have been found. As result, the use of μdeformation leads to improved agreement with observational data as compared to the wellknown models (T. Harko and NavarroFrenkWhite 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 BoseEinstein condensed dark matter, under the equilibrium condition the partition function and macroscopic characteristics are computed and compared with the results following from the GrossPitaevskii 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 ThomasFermi approximation are formulated in a unified way. The deformation of the commutation relations for the macroscopic wave function and the corresponding modification of the GrossPitaevskii 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 Bosecondensation 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 BoseEinstein, FermiDirac, classical and infinite statistics. 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 skewsymmetric classical rmatrices, we have found, in terms of the corresponding rmatrices, the sufficient conditions which guarantee that the "magic recipe of Sklyanin" indeed produces canonical coordinates. An example of skewsymmetric trigonometric rmatrix, 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 singlepoint interactions under the application of external fields has been developed. The essential feature is an asymptotic method based on the onepoint approximation of multilayered heterostructures that are subject to bias potentials. In the zerothickness limit, the matrix connecting the twosided 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 "onepoint" models of two and threeterminal 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 onepoint 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 socalled κaddition for particle energies is proposed. A κdependent Hamiltonian of the onemode multiphoton (sub)system is introduced and employed, with different modes being regarded as independent, to consider the thermodynamics of the blackbody radiation and to obtain main thermodynamic quantities using analytical tools of the meanfield approximation and numerical evaluations based on exact formulas. The entropy of a onemode 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 deltalike barrier with at least one adjacent well located at nonzero sufficiently small distance has been observed. It is shown that in the zerothickness 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 taufunction of the FujiSuzukiTsuda 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 AGTW relation for c=N−1 was obtained.N. Iorgov
 Bosecondensate models of dark matter are rather popular, however possess some difficulties. We consider the μBose 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 μBGM, the scalar curvature is calculated and its singular behavior is found. It confirms a presence of condensate in the system of μbosons. Using this fact and other important properties of μBGM which give some advantages relative to BGM, the application of the μdeformed system for effective modeling of dark matter surrounding dwarf galaxies is proposed.A. Gavrilik, I. Kachurik, M. Khelashvili, A. Nazarenko
 For the onedimensional Schrödinger equation, which describes the electron tunneling through a two or threelayer heterostructure, in a singlepoint approximation, the effect of splitting resonanttunneling 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 FujiSuzukiTsuda system with n singular points in terms of semidegenerate npoint conformal blocks of WNalgebra with central charge c=N1 is constructed. This construction uses different properties of conformal blocks of WNalgebras. 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 threepoint conformal blocks to a minimal set of conformal blocks is given; (2) for c=N1 for the degenerate field on the second level the restrictions on the fields to have nonzero threepoint conformal blocks were found; (3) formula for the fourpoint conformal blocks with one semidegenerate and one degenerate fields in terms of generalized hypergeometric functions is proved.P. Gavrylenko, N. Iorgov
 Ideal Ddimensional Bose gas is considered in a trap (external potential) in more general case than in previous works, namely, for isotropic potential U(r) = Ar^{n} +Br^{m}, where 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 BoseEinstein 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 onepoint interactions realized in the limit when the thickness of two and threelayer 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 multivalued 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 nonGaussian 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
Articles in journals, other publications
 A.M. Gavrilik, I.I. Kachurik and A.V. Nazarenko, New deformed Heisenberg algebra from the
 A.M. Gavrilik and A.V. Nazarenko, Axionlike Dark Matter Model Involving Two Phase Structure and TwoParticle Composites (Dimers). Phys. Rev. D (accepted); arXiv:2309.03290 (15 pages). https://doi.org/10.48550/arXiv.2309.03290
 A.V. Zolotaryuk, Y. Zolotaryuk, V.P. Gusynin Bound states and point interactions of the onedimensional pseudospinone Hamiltonian, J. Phys. A: Math. Theor. v. 56 (48), 485303 https://doi.org/10.1088/17518121/ad075e
 O. Gamayun, Yu. Zhuravlev, N. Iorgov, On LandauerButtiker formalism from a quantum quench, J. Phys. A: Math. Theor. v. 56 (2023), id. 205203. https://iopscience.iop.org/article/10.1088/17518121/accabf
 Yu. Bespalov, L. Kovalchuk, H. Nelasa, R. Oliynykov, R. Viglione, Models for Generation of Proof Forest in zkSNARK Based Sidechains, Cryptography, v. 7 (1) (2023), 27 pp. https://doi.org/10.3390/cryptography7010014
 T. Skrypnyk, Elliptic Gaudintype model in an external magnetic field and modified algebraic Bethe ansatz, Nuclear Physics B, vol. 988, 116102 (2023). https://doi.org/10.1016/j.nuclphysb.2023.116102
 T. Skrypnyk, Elliptic BCSRichardson model and the modified algebraic Bethe ansatz, J. Phys. A: Math. Theor. vol. 56 (2023), id. 205202. https://doi.org/10.1088/17518121/acc857
 T. Skrypnyk, Symmetric and asymmetric separated variables. II. A second integrable case of the complex Kirchhoff model, J. Math. Phys., vol. 64, 093503 (2023). https://doi.org/10.1063/5.0152882
Preprints
 N. Iorgov, Yu. Zhuravlev, Longtime asymptotic series for the Painleve II equation: Riemann–Hilbert approach, https://arxiv.org/abs/2311.17051  2023.
Articles in journals, other publications
 A.M. Gavrilik, A.V. Nazarenko, Bose–Einstein condensate dark matter that involves composites. Universe 8(3) , 187 (2022).
 V.P. Gusynin, O.O. Sobol, A.V. Zolotaryuk, Y. Zolotaryuk, Bound states of a onedimensional Dirac equation with multiple deltapotentials, Low Temperature Physics/Fizyka Nyzkykh Temperatur, V.48, No.12, pp. 1157–1168 (2022).
 O. Gamayun, N. Iorgov, Yu. Zhuravlev, Effective freefermionic form factors and the XY spin chain, SciPost Phys. 10, 070 (2021).
 Yu. Zhuravlev, E. Naichuk, N. Iorgov, and O. Gamayun, Largetime and longdistance asymptotics of the thermal correlators of the impenetrable anyonic lattice gas, Phys. Rev. B 105, 085145 (2022).
 T. Skrypnyk, Anisotropic BCSRichardson model and algebraic Bethe ansatz, Nuclear Physics B, 975, 115679 (2022).
 T. Skrypnyk, The Generalized LipkinMeshkovGlick Model and the Modified Algebraic Bethe Ansatz, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 18, 074 (2022).
 T .Skrypnyk, Anisotropic Z ngraded classical rmatrix, deformed An Todaand Gaudintype models, and separation of variables, Journal of Mathematical Physics, 63, 091702 (2022).
 T .Skrypnyk, On the general solution of the permuted classical Yang–Baxter equation and quasigraded Lie algebras, Journal of Mathematical Physics, 63, 033507 (2022).
 Y. Tykhyy, Finite orbits of monodromies of rank two Fuchsian systems, Analysis and Mathematical Physics 12, article id.122 (2022). https://doi.org/10.1007/s13324022006982
 Lyudmila Kovalchuk, Roman Oliynykov, Yuri Bespalov, Mariia Rodinko, Methods of ensuring privacy in a decentralized environment, Information Security Technologies in the Decentralized Distributed Networks,  Lecture Notes on Data Engineering and Communications Technologies, Vol. 115, SpringerVerlag, 2022, https://doi.org/10.1007/9783030951610_1, pp. 132.
 Lyudmila Kovalchuk, Roman Oliynykov, Yuri Bespalov, Mariia Rodinko, Analysis and research of threat, attacker and security models of data depersonalization in decentralized networks, Information Security Technologies in the Decentralized Distributed Networks,  Lecture Notes on Data Engineering and Communications Technologies, Vol. 115, SpringerVerlag, 2022, https://doi.org/10.1007/9783030951610_3, pp. 7188.
 Lyudmila Kovalchuk, Roman Oliynykov, Yuri Bespalov, Mariia Rodinko, Cryptographic mechanisms that ensure the efficiency of SNARKsystems, Information Security Technologies in the Decentralized Distributed Networks,  Lecture Notes on Data Engineering and Communications Technologies, Vol .115, SpringerVerlag, 2022, https://doi.org/10.1007/9783030951610_8, pp. 185201.
 Lyudmila Kovalchuk, Roman Oliynykov, Yuri Bespalov, Mariia Rodinko, Comparative analysis of consensus algorithms using a directed acyclic graph instead of a blockchain, and the construction of security estimates of spectre protocol against double spend attack, Information Security Technologies in the Decentralized Distributed Networks,  Lecture Notes on Data Engineering and Communications Technologies, Vol.115, SpringerVerlag, 2022, https://doi.org/10.1007/9783030951610_9, pp. 203224.
Papers at conferences and seminars
 Yuri Bespalov, Lyudmila Kovalchuk, Hanna Nelasa, Roman Oliynykov, and Alberto Garoffolo, Game theory analysis of incentive distribution for prompt generation of the proof tree in zkSNARK based sidechains, 2022 IEEE International Carnahan Conference on Security Technology (ICCST), 2022, https://doi.org/10.1109/ICCST52959.2022.9896484
Articles in journals, other publications
 O. M. Gavrilik and A. V. Nazarenko, Phases of the Bose–Einstein Condensate Dark Matter Model with Both Two and ThreeParticle Interactions, Universe 2021, 7(10), 359. (Q2) https://doi.org/10.3390/universe7100359
 A.V. Zolotaryuk and Y. Zolotaryuk, Scattering data and bound states of a squeezed doublelayer structure, J. Phys. A: Math. Theor. 54 (2021) 035201 (29pp) https://doi.org/10.1088/17518121/abd156
 O. Gamayun, N. Iorgov, Yu. Zhuravlev, Effective freefermionic form factors and the XY spin chain, SciPost Phys. 10, 070 (2021). doi: DOI: 10.21468/SciPostPhys.10.3.070 [Not yet assigned quartile (2020). Impact Factor (2020): 6.16]
 T. Skrypnyk, N. Manojlovic "Twisted rational rmatrices and algebraic Bethe ansatz: Application to generalized Gaudin and Richardson models", Nuclear Physics B, 967, 115424 (2021). Quartile 2020: Q1 https://doi.org/10.1016/j.nuclphysb.2021.115424
 T. Skrypnyk "Separation of Variables, QuasiTrigonometric rMatrices and Generalized Gaudin Models", SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 17, 069 (2021). Quartile 2020: Q2 https://doi.org/10.3842/SIGMA.2021.069
 T. Skrypnyk "On a class of gl (n)⊗ gl (n)valued classical rmatrices and separation of variables", Journal of Mathematical Physics, 62, 063508, (2021). Quartile 2020: Q2 https://doi.org/10.1063/5.0041967
 T. Skrypnyk "Symmetric and asymmetric separation of variables for an integrable case of the complex Kirkhoff’s probleml", Journal of Geometry and Physics, 172,104418 (2022). Quartile 2020: Q2 https://doi.org/10.1016/j.geomphys.2021.104418
 Л. Ковальчук, Р. Олійников, Ю. Беспалов та ін.; за загальною редакцією Р. Олійникова, О. Кузнецова та О. Лемешка. Технології інформаційної безпеки в децентралізованих розподілених мережах. Частина 1. – Харків: Видавництво «Форт», 2021. – 300 с. ISBN 9786176300656
 Yuri Bespalov, Alberto Garoffolo, Lyudmila Kovalchuk, Hanna Nelasa, and Roman Oliynykov, Probability models of distributed proof generation for zkSNARKbased blockchains, Mathematica 9 (2021), no. 23, 31pp, https://www.mdpi.com/22277390/9/23/3016 https://doi.org/10.3390/math9233016. Journal Rank: JCR  Q1 (Mathematics) / CiteScore  Q1 (General Mathematics)
 Yuri Bespalov, Alberto Garoffolo, Lyudmila Kovalchuk, Hanna Nelasa, and Roman Oliynykov, Gametheoretic view on decentralized proof generation in zkSNARK based sidechains, Cybersecurity Providing in Information and Telecommunication Systems (CPITS 2021),  CEUR Workshop Proceedings, vol. 2923, 2021, http://ceurws.org/Vol2923/, pp. 4759
Articles in journals, other publications
 A.M. Gavrilik, M.V. Khelashvili and A.V. Nazarenko. BoseEinstein condensate dark matter model with threeparticle interaction and twophase structure, Phys. Rev. D 102 (2020), 083510 (13pp.).
 A. Zolotaryuk, Y. Zolotaryuk. Point interactions with bound states: A zerothickness limit of a doublelayer heterostructure, Low Temperature Physics (Fizika Nizkikh Temperatur), Vol. 46, 927–933 (2020).
 A.V. Zolotaryuk and Y. Zolotaryuk. Scattering data and bound states of a squeezed doublelayer structure, Journ. of Physics A: Math. Theor. (2020), No.12, pp.134. DOI: 10.1088/17518121/abd156
 A.V. Nazarenko. Partition function of the Bose–Einstein condensate dark matter and the modified Gross–Pitaevskii equation, Int. J. Mod. Phys. D 29, No. 2 (2020) 2050018 (18 pages). DOI: 10.1142/S0218271820500182
 P. Gavrylenko, N. Iorgov, O. Lisovyy. Higher rank isomonodromic deformations and Walge¬b¬ras, Lett. Math. Phys. 110, pp. 327–364 (2020), DOI: 10.1007/s11005019012076
 T. Skrypnyk. Separation of variables for quadratic algebras: algebras of Maillet and reflectionequation algebras, J. Math. Phys., Vol. 61, issue 8 (2020), paper id. 083504.
 G. Magnano, T. Skrypnyk. New separation of variables for the classical XXX and XXZ Heisenberg models, SIGMA, Vol. 16, 047, (2020)
 T. Skrypnyk. Symmetric separation of variables for trigonometric integrable models, Nuclear Physics B, 957, 115101, (2020).
 T. Skrypnyk. Separation of variables, Laxintegrable systems and gl(2)хgl(2)valued classical rmatri¬ces, Journal of Geometry and Physics, 155, 103733, (2020).
 G. Bonelli, F. Del Monte, P. Gavrylenko, A. Tanzini. N = 2* Gauge Theory, Free Fermions on the Torus and Painlevé VI, Commun. Math. Phys. v. 377, pp. 1381–1419 (2020).
Preprints
 A.M. Gavrilik, M.V. Khelashvili and A.V. Nazarenko. BoseEinstein condensate dark matter model with threeparticle interaction and twophase structure, arXiv: 2007.04832 [astroph.GA].
 A. Gamayun, N. Iorgov, Yu. Zhuravlev. Effective freefermionic form factors and the XY spin chain, arxiv: 2012.02079.
 M. Lizunova, O. Gamayun. Solitary wave propagation in media with steplike inhomogeneuities, arXiv:2010.03385.
 A.V. Zolotaryuk and Y. Zolotaryuk. Scattering data and bound states of a squeezed doublelayer structure, arxiv:2011.11437.
 O. Gamayun, A. Slobodeniuk, J.S. Caux, O. Lychkovskiy. Nonequilibrium phase transition in transport through a driven quantum point contact, arXiv:2006.02400.
 Yu.Tykhyy. Finite orbits of monodromies of rank two Fuchsian systems, arXiv:2010.08477.
Papers at conferences and seminars
 Yu. Bespalov, L Kovalchuk, H. Nelasa, R. Oliynykov. On Generation of Cycles, Chains and Graphs of PairingFriendly Elliptic Curves, Proc. of IEEE International ScientificPractical Conf. "Problems of Infocommunications. Science and Technology" (PIC S&T), Kharkiv 2020.
Articles in journals, other publications
 W.S. Chung, A.M. Gavrilik, A.V. Nazarenko, Photon gas at the Planck Scale within the doubly special relativity, Physica A: Stat. Mech. & Its Applics. 533 (2019), paper id.121928.
 A.M. Gavrilik, A.V. Nazarenko, Statistics effects in the extremal black holes ensemble, Int. J. Mod. Phys. A 34 (2019), paper id. 1950215 (16 pages).
 A.M. Gavrilik, I.I. Kachurik, M.V. Khelashvili, Galaxy rotation curves in the μdeformationbased approach to dark matter, Ukr. J. Phys. 64 (2019), No.11, pp. 10421049.
 A.M. Gavrilik, Yu.A. Mishchenko, Composite fermions as deformed oscillators: Wavefunctions and entanglement, Ukr. J. Phys. 64 (2019), No.12, pp. 10341042.
 A.V. Zolotaryuk, G.P. Tsironis, Y. Zolotaryuk. Point interactions with bias potentials, Frontiers in Physics, 2019, vol. 7, No. 87.
 A.V. Zolotaryuk. Contact interactions in onedimensional quantum mechanics: A family of generalized δ′potentials, Ukr. J. Phys. 64 (2019), No. 11, pp. 10211028.
 P. Gavrylenko, N. Iorgov, O. Lisovyy, Higher rank isomonodromic deformations and Walgebras, Lett. Math. Phys. (2019), Online: 14 September 2019, pp. 138, https://doi.org/10.1007/s11005019012076
 Yu. Bespalov, Categories: Between Cubes and Globes. Sketch I, Ukr. J. Phys. 64 (2019), No. 12, pp. 11251128.
 Skrypnyk T. Classical rmatrices elliptic BCS and Gaudin models and spectral problem, Nuclear Physics B, 941, 225248 (2019).
 Skrypnyk T. Symmetric separation of variables in the Clebsch system, Journal of Geometry & Physics, 135 (2019), pp.204218.
 Skrypnyk T., Dubrovin B. Separation of variables for quadratic Lax algebras and skewsymmetric classical rmatrices, Journ.f Math. Phys., 60 (2019), 093506.
 M. Cafasso, P. Gavrylenko, O. Lisovyy, Tau functions as Widom constants, Commun. Math. Phys. 365 (2019), p. 741.
 M. Bershtein, P. Gavrylenko, A. Marshakov, Cluster Toda chains and Nekrasov functions, Theor. Math. Phys. 198 (2019), p. 157.
 P. Gavrylenko, R. Santachiara, Crossing invariant correlation functions at c=1 from isomonodromic τ functions, J. High Energy Phys. (2019) 2019: 119.
Preprints
 A.V. Nazarenko, Partition function of the Bose–Einstein condensate dark matter and the modified Gross–Pitaevskii equation, arXiv:1910.00232 [astroph.GA].
 G. Bonelli, F. Del Monte, P. Gavrylenko, A. Tanzini, N=2∗ gauge theory, free fermions on the torus and Painlevé VI, arXiv:1901.10497 [hepth].
 G. Bonelli, F. Del Monte, P. Gavrylenko, A. Tanzini, Circular quiver gauge theories, isomonodromic deformations and WN fermions on the torus, arXiv:1909.07990 [hepth]
Papers at conferences and seminars
 A.M. Gavrilik, I.I. Kachurik, Hamiltonians built from pseudoHermitian position/ momentum operators, and nonlinear Bogolyubov transformations, XI BolyaiGaussLobachevsky (BGL2019) International Conference: NonEuclidean, NonCommutative Geometry and Quan¬tum Physics, May 19  24, 2019, Kiev, Ukraine. Book of Abstracts, p.12.
 Khelashvili M.V., Gavrilik A.M., Kachurik I.I. Galaxy rotation curves in the μdeformation based approach. XI BolyaiGaussLobachevsky (BGL2019) International Conference: NonEuclidean, NonCommutative Geometry and Quantum Physics, May 19  24, 2019, Kiev, Ukraine. Book of Abstracts, p.19.
 A.V. Nazarenko, A.M. Gavrilik, Statistics effects in extremal black hole ensemble, XI BolyaiGaussLoba¬chevsky (BGL2019) International Conference: NonEuclidean, NonCom¬mu¬tative Geometry and Quantum Physics, May 1924, 2019, Kiev, Ukraine. Book of Abstracts, p.29.
 A.M. Gavrilik, I.I.Kachurik, New deformed Heisenberg algebra from the μdeformation based model of dark matter, Bogolyubov Kyiv Conference Problems of Theoretical and Mathematical Physics, Sept. 24 – 26, 2019, Program and Abstracts, p. 28.
 Yu.A. Mishchenko, A.M. Gavrilik, Concept of Duality of deformed analogs of Bose gas model, and its applications, Bogolyubov Kyiv Conference Problems of Theoretical and Mathematical Physics, Sept. 24 – 26, 2019, Program and Abstracts, p. 76.
 Yu.A. Mishchenko, A.M. Gavrilik, Composite quantum particles as deformed oscillators oscillators: wave¬function, entanglement and implications , XI BolyaiGaussLobachevsky (BGL2019) International Conference: NonEuclidean, NonCommutative Geometry and Quantum Physics, May 1924, 2019, Kiev, Ukraine. Book of Abstracts, p.27.
 A.V. Zolotaryuk. Contact interactions in quantum mechanics, XI BolyaiGaussLobachevsky (BGL2019) International Conference: NonEuclidean, NonCommutative Geometry and Quant.Physics, May 1924, 2019, Kiev, Ukraine. Book of Abstracts, p. 47.
 A.V. Zolotaryuk. Contact interactions in heterostructures: A squeezed limit, International Conference CHAOS 2019, 12th Chaotic Modeling and Simulation, Book of Abstracts, p. 139.
 M. Khelashvili, A. Gavrilik, I. Kachurik, Modeling of galaxy rotation curves using nonstandard (µdeformed) statistics. Intt. School of Subnucl. Phys., 57th Course: In search for the unexpected (ISSP 2019) 2130 June 2019, EMFCSC, Erice, Italy.
 M. Khelashvili, A. Gavrilik, I. Kachurik, Galaxy Rotation Curves Within µDeformed Dark Matter Model 28th Annual Student Conference: Week of Doctoral Students 2019, 46 June 2019, Charles University, Prague, Czech Republic. WDS'19 Proceedings of Contributed Papers: Part III — Physics, Prague, Matfyzpress, pp. 4849, 2019.
 Yu. Bespalov Categories: between cubes and globes., XI BolyaiGaussLobachevsky (BGL2019) International Conference: NonEuclidean, NonCommutative Geometry and Quantum Physics, May 1924, 2019, Kiev, Ukraine. Book of Abstracts, p. 7.
Articles in journals, other publications
 A.M. Gavrilik, I.I. Kacurik, M.V. Khelashvili, A.V. Nazarenko. Condensate of μBose gas as a model of dark matter, Physica A: Stat. Mech. & Its Applics. 506 (2018), pp.835843.
 A.M. Gavrilik. Geometric Aspects and Some Uses of Deformed Mo¬dels of Thermostatistics, Universe, vol. 4 (2018), paper id. 0033, 11 pages.
 A.M.Gavrilik, I.I.Kachurik. PseudoHermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscilators, Mod. Phys. Lett. A (December 2018), online DOI:https://doi.org/10.1142/S021773231950007X
 A.V. Zolotaryuk. Origin of resonant tunneling through singlepoint barriers, Physica E: Lowdimens. Systems & Nanostruct. 103 (2018), pp. 8186.
 A.V. Zolotaryuk. A phenomenon of splitting resonanttunneling onepoint interactions, Annals of Physics, 2018, vol. 396, pp. 479494.
 P. Gavrylenko, N. Iorgov, O. Lisovy. On solutions of the FujiSuzukiTsuda system, SIGMA 14 (2018), paper. id. 123, 27 pages.
 T. Skrypnyk. Modified nlevel, n1 mode TavisCummings model and algebraic Bethe ansatz, J. Phys. A: Math. Theor. 51 (2018), paper id. 015204, 12 pages.
 T. Skrypnyk. Reductions in soliton hierarchies and special points of classical rmatrices, J. Geom. & Phys. 130 (2018), pp.260287.
 T. Skrypnyk. Separation of variables in the anisotropic ShottkyFrahm model, Theor. Math. Phys. 196 (2018), pp.13591377.
 T. Skrypnyk. B. Dubrovin, Separation of variables for linear Lax algebras and classical rmatrices, J. Math. Phys. 59 (2018), paper. id. 091405, 18 pages.
 A.M. Pavlyuk. Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants, Ukr. J. Phys. 63 (2018), No. 6, pp. 488494.
 C. Marboe, D. Volin. The full spectrum of AdS5/CFT4 I: Representation theory and oneloop Qsystem, J. Phys. A 51 (2018), paper id. 165401, 53 pages.
Articles in journals, other publications
 A.V. Zolotaryuk. Families of onepoint interactions resulting from the squeezing limit of the sum of two and threedeltalike potentials. J. Phys. A: Math. Theor., 2017, 50, 225303 (19pp).
 A.V. Nazarenko, V. Blavatska. A onedimensional random walk in a multizone environment, J. Phys. A: Math. Theor., 50, 185002 (2017), 11 pp.;
 A.V. Nazarenko, V. Blavatska. Asymmetric random walk in a onedimensional multizone environment, Ukr. J. Phys., 62, № 6 (2017), pp. 508517;
 T. Skrypnyk. Separation of variables in anisotropic models and nonskewsymmetric elliptic rmatrices, Letters in Mathematical Physics, V. 107, No 5, (2017), p.793812.
 T. Skrypnyk. Separation of variables in anisotropic models: anisotropic Rabi and elliptic Gaudin model in an external magnetic field, J. Phys. A: Math. Theor., 50, 325206 (2017).
 M.A. Bershtein, A.I. Shchechkin. Backlund transformation of Painleve III (D8) tau function, J. Phys. A: Math. Theor. 50 (2017) 115205, 28 pages.
 M.A. Bershtein, A.I. Shchechkin. qdeformed Painleve tau function and qdeformed conformal blocks, J. Phys. A: Math. Theor. 50 (2017) 085202, 21 pages.
 Christian Marboe, Dmytro Volin. Fast analytic solver of rational Bethe equations, J. Phys. A: Math. Theor., 50 (2017), 204002, 12 pages.
Articles in journals, other publications
 Gavrilik A.M., Kachurik I.I., New version of pseudoHermiticity in the TwoSided Deformation of Heisenberg Algebra, Mod. Phys. Lett. A 31, no.4 (2016) paper 1650024 (14 pp.).
 Alexandre M. Gavrilik, Ivan I. Kachurik, Nonstandard Deformed Oscillators from q and p, qDeformations of Heisenberg Algebra, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications), 12 (2016) paper 047 (12 pp.).
 Nazarenko A.V., Kulinich Yu.A. Relativistic kinematics of twoparametric Riemann surface in genus two. Hadronic J. 39, 181198 (2016).
 О.В. Золотарюк, Л.С. Брижик, Я.О. Золотарюк, Солітонна динаміка в м'яких молекулярних структурах, В книзі Наукова рада з проблеми Фізика м'якої речовини (ЛьвівКиїв, 2016) с. 3443.