О существенном и дискретном спектрах операторов шредингера, соответствующих системам двух и трех одинаковых частиц на решетке

Аннотация

Actuality and demand of the theme of dissertation. Numerous scientific and applied researches conducted around the world show the following fact: throughout physics stable composite objects are usually formed by way of attractive forces, which allow the constituents to lower their energy by binding together. Repulsive forces separate particles in a free space. However, in recent years scientists have proved that in a structured environment such a periodic potential and in the absence of dissipation, stable composite objects can exist even for repulsive interactions. The Bose-Hubbard models, which have been used to describe the repulsive pairs, i.e. the Schrodinger operators on lattices is the theoretical basis for the experimental observations and applications. In this regard, the study of Schrodinger operators, associated to Hamiltonians of systems of particles moving on lattices, which appear in models of solid state physics and lattice field theory, is one of the priority areas of science. In our country in the years of independence, a great attention has been paid to scientific areas having a practical importance; in particular, a great emphasis has been placed on study of Schrodinger operators associated to Hamiltonian of a system of particles moving on integer lattices. Significant results have been achieved in finding conditions for the existence of bound states and for their number, the energy of which is located outside the essential spectrum, and also to the threshold effects of the essential spectrum for Schrodinger operators, associated to systems of two and three particles on lattices. Since the spectrum of the family of the Schrodinger operators appears quite sensitive to a change of the quasi-momentum of system, solving problems related to the spectrum of these operators, in particular, to prove the existence of bound states as well as to determine their number depending to the quasi-momentum of system, for three particle discrete Schrodinger operators is of highly importance. In this regard, the implementation of investigations in the following directions is one of the main problems: to investigate the discrete spectrum of the Schrodinger operator corresponding to a system of two identical particles (bosons or fermions) with short-range pair potentials on lattices; to establish the threshold phenomenon of the essential spectrum for these operators; to obtain an asymptotic formula for the number of eigenvalues for the three-particle Schrodinger operator associated to a system of three identical particles on the three-dimensional lattice with a short-range pair interaction; to show the existence of eigenvalues of the three-particle Schrodinger operator associated to a system of three identical particles on lattices of dimensions one and two. Many research activities carried out in the aforementioned scientific areas all around the world exhibit a great interest and motivation to the topic of dissertation. The research conducted in this thesis corresponds to the tasks specified in the Decree of the President of the Republic of Uzbekistan № PD-436 on August 7, 2006 “On measures to improve coordination and management of the development of science and technology”, No. PD-916 from July 15, 2008 “On additional measures to stimulate innovative projects and technologies” and other normative and legal acts relating to the fundamental sciences. The aim of the research is studying the essential and discrete spectrum of two and thrcc-particle Schrodinger operators associated to a system of two or three identical particles (bosons or fermions) with short-range pair potentials on lattice. The scientific novelty consists of the following: the conditions for existence of the eigenvalues outside the essential spectrum of the Schrodinger operator associated to a system of two identical particles (fermions) with a short-range potential in all dimensions of the lattice is found; the finiteness of the number of eigenvalues lying outside of the essential spectrum of the Schrodinger operator associated to a system of two identical particles (fermions) with a short-range potential on lattice is proved; the number and location of eigenvalues of Schrodinger operator associated to a system of two particles (fermions), interacting on neighboring sites of lattice for all values of the parameters of the operator is determined; the asymptotic behavior of eigenvalues lying below of the essential spectrum of the Schrodinger operator associated to a system of three particles (bosons) with short-range pair potentials in the three-dimensional lattice is studied; the finiteness of the number of eigenvalues lying below the essential spectrum of the Schrodinger operator associated to a system of three particles (bosons) with short-range pair potentials in the three-dimensional lattice for nonzero values of the quasi-momentum in the neighborhood of zero is shown; the existence of eigenvalues of the Schrodinger operator associated to a system of three particles with pair two-particle zero-range potential in the one and two-dimensional lattices is proved. Our result is the first one in the theory of the thrcc-particle Schrodinger operators. Conclusion The thesis is devoted to investigate the essential and discrete spectra of two and thrcc-particle Schrodinger operator corresponding to the system of two or three identical particles (bosons or fermions) interacting via short-range pair potentials on lattices. Basic results of the research arc as follows. 1. We introduce the notion of resonance for the Schrodinger operator corresponding to a system of two identical particles (fermions) interacting via short-range potential on one-dimensional and two-dimensional lattice. 2. We find the conditions for existence of the eigenvalues lying outside of the essential spectrum of the Schrodinger operator corresponding to a system of two identical particles (fermions) interacting via short-range potential for all dimensions of the lattice. 3. We determine the number and location of the eigenvalues of the Schrodinger operator corresponding to a system of two particles (fermions), interacting on neighboring sites of latticcfor all parameters of the operator. 4. We obtain an asymptotic formula for the number of eigenvalues lying to the left of the essential spectrum of the Schrodinger operator corresponding to a system of three particles (bosons) with short-range pair potentials in the thrccdimcnsional lattice. 5. We show the finiteness of the number of eigenvalues lyingbclow the essential spectrum of the Schrodinger operator corresponding to a system of three particles (bosons) with short-range pair potentials on thcthrcc dimensional lattice for nonzero values of the quasi-momentum in the neighborhood of zero. 6. We prove the existence of an eigenvalue lying outside of the essential spectrum of the Schrodinger operator corresponding to a system of three particles with pair contact potentials on one dimensional and two dimensional lattice, which is the only result in the theory of discrete Schrodinger operators. 7. Weestablish the finiteness of the number of eigenvalues lying below the bottom of the essential spectrum of the Schrodinger operator corresponding to a system of three particles with zero-range pair potentials on oncand two-dimensional lattices.