Non-symmetrized hyperspherical harmonic basis for $A$-bodies

摘要

The use of the hyperspherical harmonic (HH) basis in the description of boundstates in an $A$-body system composed by identical particles is normallypreceded by a symmetrization procedure in which the statistic of the system istaken into account. This preliminary step is not strictly necessary; the directuse of the HH basis is possible, even if the basis has not a well definedbehavior under particle permutations. In fact, after the diagonalization of theHamiltonian matrix, the eigenvectors reflect the symmetries present in it. Theyhave well defined symmetry under particle permutation and the identification ofthe physical states is possible, as it will be shown in specific cases. Theproblem related to the large degeneration of the basis is circumvented byconstructing the Hamiltonian matrix as a sum of products of sparse matrices.This particular representation of the Hamiltonian is well suited for anumerical iterative diagonalization, where only the action of the matrix on avector is needed. As an example we compute bound states for systems with$A=3-6$ particles interacting through a short-range central interaction. Wealso consider the case in which the potential is restricted to act in relatives-waves with and without the inclusion of the Coulomb potential. This verysimple model predicts results in qualitative good agreement with theexperimental data and it represents a first step in a project dedicated to theuse of the HH basis to describe bound and low energy scattering states in lightnuclei.