ХУРРАМОВ АБДИМАЖИД МОЛИКОВИЧ ПАНЖАРАДАГИ ИККИ ЗАРРАЧАЛИ СИСТЕМАГА МОС БАЪЗИ МОДЕЛ ОПЕРАТОРИНИНГ СПЕКТРАЛ ХОССАЛАРИ

Аннотация

OF DISSERTATION OF THE DOCTOR OF PHILOSOPHY (PhD) ON PHYSICAL AND MATHEMATICAL SCIENCES Samarkand – 2018 35 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 В2017.2.PhD/FM55. Dissertation has been prepared at Samarkand State University. 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: Lakaev Saidahmat Norjigitovich Doctor of Physical and Mathematical Sciences, Academician Official opponents: Khalmuhammedov Alimdjan Rakhimovich Doctor of Physical and Mathematical Sciences, Professor Imomkulov Sevdiyor Akramovich Doctor of Physical and Mathematical Sciences, Professor Leading organization: Karshi State University Defense will take place «____» _____________2018 at _____ at the meeting of Scientific Council number PhD.27.06.2017.FM.02.01at 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.M.Khalxujaev Scientific secretary of scientific council on award of scientific degrees, D.F.-M.S. I.A.Ikromov Acting Scientific secretary of scientific council on award of scientific degrees, D.F.-M.S. professor 36 INTRODUCTION (abstract of PhD thesis) The aim of the research work is to study the spectral properties of a model operator associated to a system of two arbitrary particles interacting via pairwise short-range attractive potentials with the special dispersion function (describing the transfer of the particles from a site to another site of a lattice). The object of the research work-model operators, associated to a system of two arbitrary quantum particles with the special dispersion function (describing the transfer of the particles from a site to another site) interacting via pairwise short-range attractive potentials Scientific novelty of the research work is as follows: ˗ It is found the conditions for (existance or absence) number of eigenvalues lying bellow the essential spectrum of a model operator with the special dispersion function, associated to a system of two arbitrary particles interacting via pairwise short-range attractive potentials on d-dimensional lattice, depending on the parameters of this operator (on interaction energy of particles and on the interaction of particles and on quasi-momentum of the system); ˗ At zero value of the quasi-momentum of the system it is found the value of the interaction energy such that the bottoms of the essential spectrum of model operators are the virtual lavel (multiple virtual lavel) or simple eigenvalue (multiple eigenvalue) when the dimension of lattice d 3,4. ˗ The conditions for being the bottom of the essential spectrum of the operator under consideration an eigenvalue (multiple eigenvalue) in d 5 dimensional case are shown; The spectral properties of the operator under consideration are studied for some compact perturbations. An invariant subspace containing the discrete spectrum of the operator under consideration is constructed. Implementation of the research results. Based on the results of the spectral properties of some model operators associated to a system of two identical particles on a lattice: The continuity of the eigenvalue of the model operator associated to the system of two arbitrary particles on the lattice are used in foreign scientific research QJ130000.2726.01K82 to describe the essential spectrum of three-particle discrete Shrödinger operator (certificate of Malaysian University of Technology, February 26, 2018). Using scientific result enabled to describe location the essential spectrum depending on the parameters of the three-particle Shrödinger operator; An invariant subspace constructed to find the number of eigenvalues of a model operator under compact perturbation associated to a system of two arbitrary quantum particles on a lattice is used to construct an invariant subspace for the three-particle discrete Schrödinger operator (certificate of Malaysian University of Technology, February 26, 2018). Using scientific result enabled to prove the finiteness of the number of eigenvalues lying outside the essential spectrum for the three-particle discrete Schrödinger operator. Existence (multiplicity) of eigenvalues of model operators associated to a 37 system of two arbitrary particles on a lattice used in foreign a