Дискретный спектр оператора Шредингера, ассоциированных с системой двух произвольных частиц на решетке

Аннотация

Actuality and demand of the theme of dissertation. Numerous scientific and applied research conducted on a global level show that everywhere in physics stable complex objects arc usually formed as a result of action of attractive forces that allow the component parts to reduce the energy in their binding. However, recent years scientists have proved that in the ordered medium stable complex objects can exist even in the case of repulsive interactions. Bose-Hubbard model is used to describe the repulsive pairs, i.e. Schrodinger operator on a lattice is the theoretical basis of experimental observations and theoretical basis for the application. Therefore, the development of research of Schrodinger operators corresponding Hamiltonians of the systems of particles on a lattice, which arc found in models of solid state physics and lattice field theory is one of the priorities. In our country in the years of independence much attention has been paid to directions of applied importance, in particular, special attention was paid to the study of Schrodinger operators corresponding to the system of particles on an integer lattice. For the Schrodinger operators significant results were achieved in determining the conditions for the existence of bound states which is located outside of the essential spectrum and for their number. Since the spectrum of the Schrodinger operators associated to the systems of two quantum particles on lattice is quite sensitive to changes in the quasi-momentum of the system, solving problems related to studies of the spectrum of the operator and to show the existence of bound states as well as determine their number plays an important role . In this regard, the implementation of targeted investigation in the following areas is one of the most important problems: investigate the discrete spectrum of the Schrodinger operator corresponding to a system of two arbitrary particles with short-range pair potentials on lattice, to establish the threshold phenomenon below the bottom or above the top essential spectrum for the operator. Research carried out in the aforementioned areas confirms the actuality of the dissertation topic. This dissertation, to some extent, serves the tasks specified in the Decrees of the President of the Republic of Uzbekistan № DP-436 dated August 7, 2006 "On Measures for Improving the Coordination and Management of the Development of Science and Technology" and № DP-916 dated July 15, 2008 "Encouraging the introduction of innovative projects and technologies in production ", № DP -2789 dated February 17, 2017 "On measures to further improve the organization, management and financing of research activities and activities of the Academy of Sciences " and № DP -4947 dated February 8, 2017 "On strategy actions for the further development of the Republic of Uzbekistan ", as well as in other normative-legal acts on this activity. The aim of the research is to study location of essential and discrete spectra as well as the number of eigenvalues of the Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact potential on lattice. The scientific novelty of the research is as follows: it is proven the existence of eigenvalue located to the right of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact repulsive (//>0) potential on d>3 dimensional lattice and it is proven regularity of corresponding eigenstate finding its exact form; it is proven that the discrete Schrodinger operator has virtual level at the right edge of essential spectrum if the dimension is J = 3,4 and it is established that the virtual state is integrable; it is defined for the fixed value of quasi-momentum the values of coupling constant which the operator has virtual level and for the fixed value of coupling constant it is separated the set of quasimomentum which the operator has eigenvalue or has not eigenvalue or has a virtual level; it is shown that the right edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5; it is proven the existence of eigenvalue located to the left of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact attractive potential on d>3 dimensional lattice and it is shown regularity of corresponding eigenstate finding its exact form it is proven that the discrete Schrodinger operator has virtual level at the left edge of essential spectrum if the dimension is d = 3,4 and it is established that the virtual state is integrable; it is shown that the left edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5. Conclusion The dissertation is devoted to study essential and discrete spectra of the two particle Schrodinger operator corresponding to system of two arbitrary particles on lattice interacting via a pair contact potential. The main results of the research arc as follows: 1. It is proven the existence of eigenvalue above the top of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact repulsive (/z > 0) potential on d > 3 dimensional lattice and it is shown the corresponding eigenstate in coordinate representation exponentially decreases at infinity; 2. It is proven that the discrete Schrodinger operator has virtual level at the right edge of essential spectrum if the dimension is J = 3,4 and it is established that the virtual state approaches to zero at infinity; 3. It is defined for the fixed value of quasi-momentum the set of the values of coupling constant which the operator has virtual level and for the fixed value of coupling constant the set of quasimomentum; 4. It is shown that the right edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5; 5. It is proven the existence of eigenvalue below the bottom of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact attractive (// > 0) potential on 4>3 dimensional lattice and it is shown that the corresponding eigenstate in coordinate representation exponentially decreases at infinity; 6. It is proven that the discrete Schrodinger operator has virtual level at the left edge of essential spectrum if the dimension is <7 = 3,4 and it is established that the virtual state approaches to zero at infinity; 7. It is shown that the left edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5;