On the Method of N-Body Hyperspherical Basis Symmentrization
This paper discusses the problem of symmetrization of N-body hyperspherical functions. In the next section, parentage scheme of symmetrization is applied to the construction of the symmetrized five- and six- particle hyperspherical basis in the (4+1) and (5+1) configurations. The relations between parentage coefficients and the KRC of the corresponding hyperspherical functions under particle permutations are obtained. It is demonstrated that the KRC contain only permutations related with the last two particles. In the third section parentage scheme of symmetrization is generalized to construct N-particle symmetrized hyperspherical functions
Parentage Scheme of Summarization (PSS) to the N-body symmetrized basis construction necessary for the description of the structural characteristics and decay reactions of the hypernuclear and nuclear systems with arbitrary amount of particles is introduced. Proposed method allows to construct N-particle symmetrized hyperspherical functions on the bases of N-particle hyperspherical functions symmetrized with respect to N–1-particles by the use of the Kinematic Rotation Coefficients (KRC) related with the (N-1)-th and N-th particle permutations. The main problem that arises when investigating dynamics of few-body systems in physics is the problem of kinematic rotations under particle permutations. When number of particles increases, kinematic rotations include not only particle permutation but also transitions between different possible configurations, and mathematical calculations using complex general formula become impossible. Moreover, no general formula exists for systems with more than four particles. In order to solve kinematic rotation problem for N-particle systems, the recurrence method of determination of the KRC is applied. According to this method, the initial coefficients with the lowest quantum numbers are calculated by solving the overlap integral analytically, wave functions with arbitrary quantum numbers are expanded in terms of the basic hyperspherical functions, and the kinematic rotations of the obtained expansion are performed with the use of already known coefficients with the lowest quantum numbers. Significant advantage of the recurrence method is that no principal difficulties arise when increasing number of particles. Furthermore, recurrence relations contain numerical coefficients that are easy to evaluate by substituting appropriate quantum numbers.
The problem of symmetrization of N-Body hyperspherical functions is solved by the use of the PSS. A construction scheme is given for the symmetrized hyperspherical basis in (4+1) and (5+1) configurations. The relations between parentage coefficients and the KRC of the basic hyperspherical functions are obtained. Parentage coefficients for N-Body systems are introduced. It is demonstrated that no principal difficulties arise when increasing number of particles.
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