## 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.

Department's staff:

chief research fellow V.O. MarchenkoDr.Sc. ,Profchief research fellow V.P. Kotlyarov Dr.Sc. leading research fellow 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. senior researcher E. Melamedova Ph.D. research fellow M.S. Filipkovska Ph.D. junior research fellow V. Vengerovsky Ph.D. junior research fellow O.O. Gukalov Ph.D. junior research fellow K. Andreiev Ph.D. engeneer E.V. Afanasjev 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.

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;

the finite-zonal almost periodic solutions of non-linear Schrodinger equations, sine-Gordon 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;

the theory of decay of step-like type solutions and non-linear integrable equations on 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.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, Berlin: Springer, Lecture Notes in Mathematics, 1987.

V.A. Marchenko, Non-linear equations and operator algebras, Kiev, Nauk.Dumka, 1986; English translation: D.Riedel, Dordrecht, 1987.

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.

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)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, 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)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.