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arXiv:nucl-th/9606022

Few-Body Quantum Problem in the Boundary-Condition Model

Alexander Konstantinovich Motovilov

Published Jun 12, 1996• 0 citations
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Abstract

Systems of three and four quantum particles in the boundary-condition model are considered. The Faddeev-Yakubovsky approach is applied to construct the Fredholm-type integral equations for these systems in framework of the Potential theory. The boundary-value problems are formulated for the Faddeev-Yakubovsky components of wave functions.

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Systems of three and four quantum particles in the boundary-condition model are considered. The Faddeev-Yakubovsky approach is applied to construct the Fredholm-type integral equations for these systems in framework of the Potential theory. The boundary-value problems are formulated for the Faddeev-Yakubovsky components of wave functions.
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.
Cluster expansion of the many-body Schr\"odinger wave function is manifested through the projected Faddeev-Yakubovsky equations. It is shown that using these equations one can formulate a systematic expansion for the wave function of finite systems in a translational invariant manner. The solution of the N-body problem at a given order becomes equivalent to solving the few-body Faddeev-Yakubovsky equations. The first two orders in this expansion are given explicitly and then solved for the case of N bosons interacting through the nucleon-nucleon Afnan-Tang S3 and Malfliet-Tjon MT-V potentials.