The Samarkand branch was established in accordance with the Decision of the President of the Republic of Uzbekistan dated May 7, 2020 No PP-4708 "On measures to improve the quality of education and development of research in the field of mathematics."

The Samarkand branch of the Institute conducts research in the following areas:

- Friedrichs models and spectral theory of Schrödinger operators on a lattice.

- Oscillation integrals and their applications.

- Inverse problems of spectral analysis and application of the document to nonlinear evaluation equations.

- Mathematical modeling of leakage of heterogeneous formations in porous media.

- Hardy-type inequalities and their applications.

At present, the Samarkand branch of the Institute conducts research in the following areas:

1.       Friedrichs models and spectral theory of Schrödinger operators on a lattice

The results in this direction were obtained by academicians S.N. Lakaev, J. Abdullaev, A.M. Khalkhozhaev, M. Muminov, Z. Muminov, Sh. Alladustov, A. Boltaev, Sh. Khamidov.

2.      Oscillation integrals and their applications.

 The results in this direction were obtained by I.A. Ikromov, A. Safarov.

3.       The inverse problems of spectral analysis and their application to nonlinear evolutionary equations.

The results in this direction were obtained by A.Hasanov.

4.       Problems of mathematical modeling of heterogeneous fluid filtration in porous media are being studied by B.Khuzhayorov and his students.

5.       Hardy-type inequalities and their applications.

 The results in this direction were obtained by K.Kuliev.

1.      Friedrichs models and spectral theory of Schrödinger operators on a lattice

·         For the Schrödinger operator corresponding to a system of three arbitrary particles (two bosons and another particle, respectively) that interact with a pair of one- and two-dimensional laticces using contact potentials, it is proved that there is a three-particle bound state at certain values of the impact constant. In particular, it has been shown that at some values of the interaction energy, there are cases where the important spectrum is connected below the lower threshold or above the upper threshold.

·      A generalized Friedrichs model with the same color of motion was studied. Its important spectrum has been described. Conditions for the existence of a eigenvalue outside the essential spectrum were found. The number and location of eigenvalues were determined, and approximations were found for eigenvalues.

·      The family of discrete Schrödinger operators, whose potential depends on two parameters consisting of a delta function and a perturbation operator, is considered in a d-dimensional lattice. The existence of eigenvalues was proved, and the existence of boundary eigenvalues and resonances, as well as their dependence on the given parameters and the size of the lattice, were clearly shown.

2.      Oscillation integrals and their applications.

·         Estimates of the Fourier transform of charges (measures) concentrated on smooth hypersurfaces are considered. Following M. Sugumoto, three classes of smooth hypersurfaces are defined. Depending on the class, estimates of the Fourier transform of charges are obtained in terms of Randol maximal functions. The obtained estimates are applied to the solution of the integrability problem for the Fourier transform of measures concentrated on some nonconvex hypersurfaces. The sharpness of the obtained estimates is shown. Moreover, The problem posed by Kevrikidis-Stefanov on the discrete Klein-Gordon was solved.

3.       The inverse problems of spectral analysis and their application to nonlinear evolutionary equations.

·        The non-oscillation conditions of the Sturm-Liouville problem were created using Hardy's inequalities. Using the Carleman function, the harmonic function and its derivatives were reconstructed to a given value at a part of the boundary of the field.  For elliptic equations, the solution of the Cauchy problem and the regulation of the product of the solution were constructed, and the stability value was obtained in the classical sense. It has been shown that the efficient construction of the Carleman function is equivalent to the construction of a regulated solution of the Cauchy problem.

4.      Hardy-type inequalities and their applications.

·           There are obtained conditions for boundedness of Hardy-Volterra type operators acting in weighted Lebeg spaces.

The members of the department cooperate with scientists from Great Britain, Italy, France, Germany, USA, Russia, Belgium, England, Turkey, Brazil, Armenia, China, Austria, Czech Republic, Australia, Malaysia, Japan, including:

1. Ruhr, Bonn, Mainz, Braunschweig of Germany, Zussex University of England, SISSA, ICTP Research Centers in Italy and the University of Rome, Columbia (Missouri) and Davis (California) Universities in the USA, MSU, Russian Information Problems Research Institute and Nuclear Research Institute in Dubna, Scientific cooperation has been established between Malaysia and Mara Technology, the group, Putra and Kebangsaan, Karlovy Vary University in Czechoslovakia and a number of other research centers.

2. S.N. Lakaev and his students in collaboration with Professor S. Albeverio of the University of Bonn (Germany) on the basis of projects DFG 436 UZB 113/3, DFG 436 UZB 113/4, DFG 436 UZB 113/6, DFG 436 UZB 113/7  from 2001 to 2011, a total of 60 months of research was conducted.

3. From 2013 to 2014, S.N. Lakaev spent 10 months at Fulbright Grant Universities in Davis, California and Columbia, Missouri. J. Mendez (Puerto Rico) gave scientific lectures and discussed research for students and scholars at university seminars.

4. Prof. I. Ikromov at the suggestion of Professor D. Muller, University of Kiel (Germany) German DFG Scientific Foundation DFG-Grant MU 761 / 11-1 (Fragen der Harmonischen Analysis im Zusammenhang mit Hyperflachen) is conducting research jointly on the basis of the project.

5. Prof. I. Ikromov has collaborated with J.C. Cuenin, a scientist at Loughborough University in the United Kingdom is currently co-authoring scientific papers with the scientist.

6. A. Safarov is scheduled to go to the University of Ghent, Belgium for a two month internship (01.10.2021 -30.11.2021) through the "El Yurt Umidi" Foundation. It is planned to establish scientific cooperation with Professor M. Ruzhansky.

7. Regular contacts have been established with compatriots abroad, including A. Khalkhozhaev, M. Pardaboev, in collaboration with Professor of the University of Vienna, Austria Sh. Kholmatov, 2 articles have been sent to foreign journals.

8. K. Kuliev has established scientific cooperation with scientists of the Czech Republic P.Drabek and A. Kufner.

9. Scientific cooperation has been established with Sh. Alladustov and professors of Curtin University of Australia A. Kadirov and I. Bray.

10. Scientific cooperation has been established with Z. Muminov and UCIM of Malaysia, professors of Kyushe University of Japan A. Narzullaev, Z. Eshquvvatov, F. Hiroshima.