Figure 1

Magnitude of the matrix elements without [panels (a) and (b)] and with [panels (a${}^{\prime }$) and (b${}^{\prime }$)], which sort the diagonal matrix elements from the smaller to the larger. The color from blue to red corresponds to values from zero to large magnitudes. The results are based on $I^{\pi }=0^{+}$ [panels (a) and (a′)] and $I^{\pi }=2^{+}$ [panels (b) and (b${}^{\prime }$)] states of the ${}^{24}\mathrm{Mg}$ nucleus, obtained by using the USD interactions. The magnitude of $H_{\mathit{ij}}^{(I)}$ without sorting the diagonal matrix elements (left-hand side) are close to random, and those with sorting the diagonal matrix elements (right-hand side) decrease rapidly as going farther from the diagonal line.Reuse & Permissions

Figure 2

Distribution for $H_{\mathit{ij}}^{(I)}$ to be nonzero, denoted by $\rho (d)$, and the lowest eigenvalues of $h$, denoted by ${\epsilon }_d(h)$. The results of (a), (a′) and (b), (b′) are obtained for the $J^{\pi }=0^{+}$ and $J^{\pi }=2^{+}$ states of ${}^{24}\mathrm{Mg}$, respectively. One sees that the values of $d$ where $\rho (d_0)=0$ and where the ${\epsilon }_d(h)\text{−}\ln \hspace{0.5em}d$ plots change the slope ($D_0$) coincide (i.e., $d_0=D_0$). We use the same scale in (a), (a′) or (b), (b′). The dotted lines are used to guide the eye.Reuse & Permissions

Figure 3

Distribution for $\ln (D_0)/\ln (d_0)$ and $k^{\prime }/k$ for 200 $H^{(I)}$ matrices for a number of nuclei in the $\mathit{sd}$ shell, the dimension $D$ of which goes from 500 to 10 000, by taking the USD interactions. One sees that $\ln (D_0)/\ln (d_0)\simeq 1$, and that $k^{\prime }~\frac{2}{5}k$ with considerable fluctuations. In this paper, we assume that $\ln (D_0)/\ln (d_0)=1$ and $k^{\prime }=\frac{2}{5}k$ when we predict the lowest eigenvalue of $H^{(I)}$, for simplicity.Reuse & Permissions

Figure 4

Comparison of low-lying levels obtained by exact diagonalization (denoted by exact) and those predicted by using Eq. (1) (denoted by pred1) and the linear correlation formula suggested in Ref. [8] (denoted by pred2) for the ${}^{24}\mathrm{Mg}$ nucleus in panel (a), and the ${}^{28}\mathrm{Si}$ nucleus in panel (b). The results are obtained by using the Yukawa-type interaction of Refs. [10, 13]. One sees substantial improvement of predictions for the lowest eigenvalues for each spin $I$ by using Eq. (1) of this paper.Reuse & Permissions