of the National Academy of Sciences of Ukraine

## DEPARTMENT OF MATHEMATICAL PHYSICS

Head of the Department: Doctor of Sciences

SHCHERBINAMariya Vladimirovnatel: +(380)-57-330-85-85

fax: +(38)-057-340-33-70

E-mail:Department of Mathematical Physics was founded in 1960. It was headed by Professor

V.O. Marchenkofrom the very beginning till 2002, from 2002 till 2016 by V.P.Kotlyarov. In different years V.G. Drinfeld, L.A. Pastur, E.Ya. Khruslov, F.S.Rofe-Beketov, V.Ya. Golodets, V.O. Shcherbina, G.M. Gestrin, D.Sh. Lundina, A.S. Sokhin, V.O. Kozel, R.M. Davydov, L.L. Vaksman, S.V. Neshveev, D.L. Shklyarov, K.S. Medynets, M.O. Kudryavtsev, O.O. Bershtein, O.O. Minakov,O.M. Karpelworked at the Department. Since 2016, the department is headed by M.V. Shcherbina

Department's staff:

chief research fellow V.O. MarchenkoDr.Sc. ,Profchief research fellow V.P. Kotlyarov Dr.Sc., Prof leading research fellow A.A. Zvyagin Dr.Sc. ,Prof.leading research fellow Yu. Freiman Dr.Sc. leading research fellow I. Egorova Dr.Sc. leading research fellow D. Shepelsky Dr.Sc. research fellow M.S. Filipkovska Ph.D. research fellow V. Vengerovsky Ph.D. research fellow O. Hukalov Ph.D. Fields of interest:

- direct and inverse problems in scattering theory and spectral analysis;
- Riemann–Hilbert problems and completely integrable nonlinear equations;
- theory of random matrices, including beta-matrix models, unitary matrix models, ensembles of sparse and diluted random matrices, band random matrices and random graphs;
- spectral theory of the differential and finite difference operators with random and almost periodic coefficients, including inverse scattering theory problems for these operators;
- integrable nonlinear partial differential and difference equations. Direct and inverse problems of spectral analysis with applications to nonlinear integrable systems. Large time asymptotic analysis for nonlinear integrable systems;
- exactly and asymptotically exactly solvable models of large interacting systems (including disordered ones), theory of phase transitions for these models and methods of the analysis of their physical characteristics.
- Inverse scattering transform method and asymptotics of solutions of nonlinear integrable peakon-type equations (Cammassa-Holm equation, Degasperis-Procesi equation, etc.)
- Riemann-Hilbert problem formalism for nonlocal nonlinear integrable equations
- Applications of inverse scattering transform methods in optical communication theory
- Theory of integrable quantum many-body low-dimensional models of condensed matter physics
- Quantum theory of magnetism and correlated systems
- Theory of quantum computers
Main results of the Department:

- the inverse scattering problem, the inverse spectral problem for the Schrodinger equation with periodic potential;
- the method for solving the wide class of problems of electromagnetic wave diffraction on periodic structures;
- the theory of averaging of problems of mathematical physics in the domains of complicated microstructure;
- Finite-gap quasi-periodic solutions for nonlinear Schrodinger and sine-Gordon equations and isotropic Heisenberg magnet;
- the new approach to constructive solving the inverse spectral problems for the differential operators with non-decreasing coefficients and its applications to constructing new classes of solutions of nonlinear evolution equations;
- Theory of splitting of steplike solutions to nonlinear integrable equations into asymptotic solitons;
- the generalization of the Riemann-Hilbert problem for non-linear equations with the step-like initial data on the whole axis and with periodic boundary conditions on the semi-axis; in particular, the algorithm for solving the mixed problem for non-linear Maxwell-Bloch equations;
- the method of asymptotic analysis of the Riemann-Hilbert problems and asymptotic behavior of corresponding solutions of non-linear equations;
- the quantum group theory and its applications to the problems of quantum field theory and geometry;
- new field of investigations in algebraic number theory, D-modules and conformal field theory;
- theory of completely integrable differential equations related to affine Lie algebras;
- the Poisson-Lie group theory and the classification of solutions of the classical Yang-Baxter equation;
- the theory of cocycles of dynamical systems and their applications to classification of different group actions on the space with measure;
- the complete description of aperiodic substitution dynamical systems in terms of Bratteli diagrams, the description of all probability ergodic invariant measures for such systems;
- the structure of the Neumann algebras and their automorphisms and its application to studying the non-commutative entropy in the statistical physics models;
- the classification of admissible representations of infinite dimensional analogues of classical matrix groups;
- the complete list of KMS states on the infinite symmetric group, invariant with respect to Young subgroups;
- the complete list of structures of
U(sl_{q}_{2}) – modular algebra on quantum plane.- Proof of the Dyson universality conjecture for the local eigenvalue statistics of beta-matrix models.
- New method of proving Central Limit Theorem for linear eigenvalue statistics of large random matrices
- Scattering theory for differential and difference operators with steplike asymptotically finite gap coefficients; integration of nonlinear equations related to these operators.
- Development of the inverse scattering transform method and its applicatio to the study of the large-time asymptotics of solutions of initial and initial-boundary value problems for the Camassa-Holm equation

and other nonlinear integrable peakon-type equations ( Degasperis-Procesi equation, Vakhnenko-Ostrovsky equation, etc.)- Theory of new classes of integrable models of quantum many-body spin and strongly correlated electron systems - the class of the multi-chain models, and the class of models with the finite concentration of magnetic impurities.
- Theory of static characteristics (the specific heat, magnetic and charge susceptibility, persistent currents, etc.) based on exact quantum solutions for quantum models of strongly correlated electron and spin models (including the ones on geometrically frustrated lattices). Conformal field theory for the asymptotic behavior of various correlation functions of such systems.
- Exact quantum theory of dynamical characteristics of many-body quantum systems of the quantum spin liquid type (spin one- and two-dimensional models, models of correlated fermions) as the result of the response to the ac external field and to sudden changes of the parameters (like pulses of the field).
- Theory for exact characteristics of systems of qubits, important for quantum computers (in particular, for topological quantum computers)
- Theories, which describe features of low-temperature experiments (the specific heat, magnetization, susceptibility, electron and nuclear magnetic resonances, muon spin rotation, Moessbouer effect, magneto-acoustic and optical experiments, etc.) on multiferroics, frustrated magnets (including spin ice ones), low dimensional magnets, molecular magnets, rare-earth and uranium compounds, heavy fermion systems, ultracold atoms on optical lattices, etc.
The main results of the Department were published in the books:

- Z.S. Agranovich and V.A. Marchenko, Inverse scattering theory, Kharkov University,1960.
- V.A. Marchenko, Spectral theory of the Sturm-Liouville operators, Kiev, Nauk.Dumka, 1972.
- V.A. Marchenko and E.Ya. Khruslov, Boundary value problems in domains with fine-grain boundary, Kiev, Nauk.Dumka, 1974.
- V.A. Marchenko, Sturm-Liouville operators and applications, Kiev, Nauk.Dumka, 1977; English translation: Birkhauser Verlag, 1985.
- L.L.Vaksman and M.S. Lifshits, Nonselfadjoint commuting operators in Hilbert space, BerlinV.A. Marchenko, Non-linear equations and operator algebras, Kiev, Nauk.Dumka, 1986; English translation: D.Riedel, Dordrecht, 1987.
- V.A. Marchenko, Non-linear equations and operator algebras, Kiev, Nauk.Dumka, 1986; English translation: D.Riedel, Dordrecht, 1987.
- Rofe-Beketov F.S. and Kholkin A.M. Spectral analysis of differential operators. Relation between spectral and oscillation properties, Mariupol, PSTU, 2001, 332pp.(russian)
- V.A. Marchenko and E.Ya. Khruslov. Homogenized models of microinhomogeneous media. Kiev, Naukova Dumka 2005.
- V.A. Marchenko. Introduction to a theory of inverse spectral analysis. Acta, Kharkov, 2005.
- Rofe-Beketov F.S. and Kholkin A.M. Spectral analysis of differential operators. Interplay between spectral and oscillatory properties. WSPC, Singapore, 2005, 461 pp.
- V.A. Marchenko, E.Ya. Khruslov, Homogenization of Partial Differential Equations. Progress in Mathematical Physics, Birkhäuser Boston, Inc., Boston, MA (2006). 398 pp.
- L. L. Vaksman Quantum bounded symmetric domains, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, V. 238, 2010, xii+256 pp.
- V.A. Marchenko Sturm–Liouville Operators and Their Applications: Revised Edition. AMS, 2011, 393 pp.
- L. Pastur, M. Shcherbina. Eigenvalue Distribution of Large Random Matrices. Mathematical Survives and Monographs, V171, American Mathematical Society: Providence, Rhode Island (2011)
- A.A. Zvyagin, Finite Size Effects in Correlated Electron Models: Exact Results, Imperial College Press, London, and World Scientific, Singapore, (2005) ISBN 978-1-86094-503-8; 978-1-78326-047-8 (ebook)
- A.A. Zvyagin, Finite Size Effects in Correlated Electron Models: Exact Results, Peking University Press, Beijing (Peking), (2012) ISBN 978-7-301-21554-8
- A.A. Zvyagin, Quantum Theory of One-Dimensional Spin Systems, Cambridge Scientific Publishers, Cambridge, (2010) ISBN 978-1-904868-85-9
The works of the Department are well known in the world, that was testified by the

Lenin Prize

(V.O. Marchenko, 1962),

Prize of the Academy of Sciences of the Soviet Union

(

V.G. Drinfeld, 1988)Fields Gold Medal

(

V.G. Drinfeld, 1990)State Prize of Ukraine

(V.O. Marchenko,1989),

(S.I. Bezuglyi, 2010)(A.A. Zvyagin, 2015)

(M.V.Shcherbina, 2018) V.I. Vernadsky Gold Medal of NAS of Ukraine

(V.O. Marchenko, 2010)

Gold Medal and Prize of Shevchenko Scientific Society (USA) and U.S.-Ukraine Foundation

(O.O. Bershtein, 2011),

(O.M. Karpel, 2013)

N.N. Krylov's Prize of the Academy of Sciences of Ukraine

(V.O. Marchenko, 1982),

(V.P. Kotlyarov,1996)N.N. Bogolyubov Prize of NAS of Ukraine

(V.O. Marchenko, 1996)

M.O. Lavrentyev Prize of NAS of Ukraine

(V.O. Marchenko, 2007)

M.V. Ostrogradsky Prize of NAS of Ukraine

(F.S. Rofe-Beketov, 2007),

(M.V. Shcherbina, 2009),

(V.P. Kotlyarov, D. Shepelsky, 2011)Prize of NAS of Ukraine for young scientists for the best scientific works

(V.M. Kulagin, 2006),

(K.S. Medynets, 2006),

(O.O. Bershtein, 2010)S.I. Pekar Prize of NAS of Ukraine

(A.A.Zvyagin, 2010)

Y.O. Mytropol's'koho Prize of NAS of Ukraine

(I.E. Egorova, 2013)

International collaboration:Université Paris-7 D.Diderot, France; University of Torun, Poland; Trondheim University, Norway; University of New South Wales, Australia; University of Ottawa, Canada; University of Copenhagen, Denmark; E. Schrödinger Institute, Vienna, Austria; University of Leipzig, Germany; University of Maryland, USA; Uppsala University, Sweden.The co-workers of the Department participate in international scientific projects such as the long-time International Science Foundation project, INTAS projects of European Community, CRDF and STCU.