Nonsymmetrized hyperspherical harmonic basis for an $A$-body system

The use of the hyperspherical harmonic (HH) basis in the description of bound states in an $A$-body system composed of identical particles is normally preceded by a symmetrization procedure in which the statistic of the system is taken into account. This preliminary step is not strictly necessary; the direct use of the HH basis is possible, even if the basis does not have a well-defined behavior under particle permutations. In fact, after the diagonalization of the Hamiltonian matrix, the eigenvectors reflect the symmetries present in it. They have well-defined symmetry under particle permutation and identification of the physical states is possible, as shown here in specific cases. The problem related to the large degeneration of the basis is circumvented by constructing the Hamiltonian matrix as a sum of products of sparse matrices. This particular representation of the Hamiltonian is well suited for a numerical iterative diagonalization, where only the action of the matrix on a vector is needed. As an example we compute bound states for systems with $A=3$–6 particles interacting through a short-range central interaction. We also consider the case in which the potential is restricted to act in relative $s$ waves with and without the inclusion of the Coulomb potential. This very simple model predicts results in qualitative good agreement with the experimental data and it represents the first step in a project dedicated to the use of the HH basis to describe bound and low-energy scattering states in light nuclei.

Figure 1

Hyperspherical tree corresponding to Eq. (7).

Figure 2

Calculated levels for $A=2$–6 using the all-wave Volkov potential.

Figure 3

Calculated levels for $A=2$, 3, 4, and 6, using the $s$-wave Volkov potential with the inclusion of the Coulomb interaction for He isotopes. In this case the $A=5$ system is unbounded.

Figure 4

$A=5$, $L=0$ levels listed in Table , denoted $E_0$, $E_1$, and $E_2$, and the $L=1$ levels listed in Table , denoted “all waves” and “$s$ wave,” normalized to the respective minimal energy $E_{\text{min}}$, are shown as functions of the nonlinear parameter $\beta$, at $K_{\max }=16$ ($L=0$ levels) and $K_{\max }=17$ ($L=1$ levels).