OF DISSERTATION OF THE DOCTOR OF PHILOSOPHY (PhD) ON PHYSICAL AND MATHEMATICAL SCIENCES Samarkand -2018 The theme of dissertation of doctor of philosophy (PhD) on physical and mathematical sciences was registered at the Supreme Attestation Commission at the Cabinet of Ministers of the Republic of Uzbekistan under number B2017.4.PhD/FM148

Аннотация

OF DISSERTATION OF THE DOCTOR OF PHILOSOPHY (PhD) ON PHYSICAL AND MATHEMATICAL SCIENCES Samarkand -2018 The theme of dissertation of doctor of philosophy (PhD) on physical and mathematical sciences was registered at the Supreme Attestation Commission at the Cabinet of Ministers of the Republic of Uzbekistan under number B2017.4.PhD/FM148. Dissertation has been prepared at Samarkand State University and Tashkent Institute of Agricultural Irrigation and Mechanization engineers The abstract of the dissertation is posted in three languages (uzbek, russian, english (resume)) on the website (www.samdu.uz) and the «Ziyonet» Information and educational portal (www.ziyonet.uz). Scientific supervisor: Xalxujaev Axmad Miyassarovich Doctor of Physical and Mathematical Sciences Official opponents: Imomkulov Sevdiyor Akramovich Doctor of Physical and Mathematical Sciences Botirov G’olib Isroilovich Candidate of Physical and Mathematical Sciences Leading organization: National university of Uzbekistan Defense will take place «____» _____________2018 at _____ at the meeting of Scientific Council number PhD.27.06.2017.FM.02.01 at Samarkand State University. (Address: University Boulevard 15, Samarkand, 140104, Uzbekistan, Ph.: (+99866) 231-06-32, fax: (+99866) 235-19-38, e-mail: patent@samdu.uz). Dissertation is possible to review in Information-resource centre at Samarkand State University (is registered №____) (Address: University Boulevard 15, Samarkand, 140104, Uzbekistan, Ph.: (+99866) 231-06-32). Abstract of dissertation sent out on «____» _______________2018 year A.S. Soleev Chairman of scientific council on award of scientific degrees, D.F.-M.S., professor A. Khatamov Acting Scientific secretary of scientific council on award of scientific degrees, D.F.-M.S. I.A. Ikromov Vice-chairman of scientific Seminar under Scientific Council on award of scientific degrees, D.F.-M.S., professor INTRODUCTION (abstract of PhD thesis) The aim of the research work is to study the existence of eigenvalues and also to obtain a convergent expansion for the eigenvalue lying outside the essential spectrum of the Schrӧdinger operator corresponding to system of two identical particles (fermions or bosons) interacting via pairwise short-range potentials on one and two dimensional lattices. The object of the research work Schrӧdinger operators corresponding to a system of two identical particles interacting via pairwise short-range potentials on one and two dimensional lattices. Scientific novelty of the research work is as follows: the existence of eigenvalue and explicit form of the corresponding eigenfunction of the two-particle Schrӧdinger operator associated to a system of two identical particles(bosons) interacting via contact attractive or repulsive potential on one and two dimensional lattices is established; a convergent expansion for eigenvalue at the coupling constant threshold of the two-particle Schrӧdinger operator associated to a system of two identical particles (bosons) interacting via contact attractive or repulsive potential on one and two dimensional lattices; it is shown that the left threshold of the essential spectrum may be a virtual level (resonance) or an eigenvalue for the two-particle Schrӧdinger operator associated to a system of two particles (fermions) interacting at neighboring sites on one-dimensional lattice and the existence or absence of eigenvalue lying to the left of the essential spectrum is proved; a convergent expansion at the coupling constant threshold and quasi-momentum threshold for the eigenvalue of the two-particle Schrӧdinger operator associated to the two-particle system (fermions) with pair interactions at neighboring sites on a one-dimensional lattice is found. Implementation of the research results. The results obtained in the dissertation are applied in the following areas: the existence of eigenvalues and threshold resonances of the two-particle discrete Schrödinger operators in the PhD dissertation of Shuhrat Lakaev with the title "Eigenvalues and resonances of the Schrödinger operators of a system of two-particles on a lattices" have been used in the research project LRGS/ TD/2011/ UKM/ ICT/03/02 (certificate of National University of Malaysia, December 28, 2017). The results and methods in the dissertation enabled to construct example for two-particle discrete Schrodinger operators having the threshold resonance, or threshold eigenvalues or regular points; the existence of eigenvalues and threshold resonance of two-particle discrete Schrödinger operators in the PhD dissertation of Shuhrat Lakaev with title "Eigenvalues and resonances of Schrödinger operators of two-particle systems on a lattice" have been used in the research project PPP/GP/FST/30/14915(certificate of Islamic Science University of Malaysia, December 27, 2017). The results and methods in the dissertation enabled to obtain the basic properties of integral operators and lattice Schrödinger operators, and also approximati