Convergence in function of the number of HO shells

In Section 5.1, we showed our strength functions calculated with $25$ HO shells. This was found to be satisfactory, and using more shells did not appreciably change the obtained strength functions. The only effect of using more oscillator shells was that we needed to use slightly more Arnoldi iterations to produce well converged results. In the case of $40$ shells, about twenty more iterations were needed, compared to calculations made with $25$ shells. In Figs. 7, 8, and 9, we show the convergence of strength functions for the $0^+$, $2^+$, and $1^-$ modes, respectively. Each panel shows the difference of two strength functions obtained in the intervals of $\Delta N_0=4$ HO shells, between $N_0$ of 22 and 38.

Figure 7: Similar as Fig. 2 but for the convergence of the $0^+$ strength functions as a function of the number of HO shells $N_{0}$.

Figure 8: Similar as Fig. 4 but for the convergence of the $2^+$ strength functions as a function of the number of HO shells $N_{0}$.

Figure 9: Similar as Fig. 6 but for the convergence of the $1^-$ strength functions as a function of the number of HO shells $N_{0}$.

These plots overstress the variations of strength functions in the sense that slight shifts of peaks create the oscillating patterns in the difference plots. To illustrate this point, in Fig. 10 we show the $1^-$ strength functions calculated for $N_0=22$, 26, 30, 34, and 38 HO shells. Poor convergence of the IS surface mode creates some uncertainty in the position and width of the high-energy bump. Larger bases should probably be used if converged results for this particular mode were required.

Figure 10: Similar as Fig. 5 but for the $1^-$ strength functions calculated for the numbers of HO shells $N_0=22$, 26, 30, 34, and 38

As noted in Ref. [5], well before the maximum number of iterations (equal to the RPA dimension $D$) is reached, the iteratively generated RPA matrix in the Krylov space can become singular. In that case, the stabilized iteration method of Eqs. (12)-(19) still protects us from obtaining complex RPA eigenvalues, but the condition number of the Krylov-space RPA matrix approaches infinity, because one or more of its eigenvalues collapse nearly to zero.

In the standard method, the RPA matrix is calculated by using the bare p-h basis states. In our method, we instead start from the pivot vector of Eq. (20) and the Arnoldi iteration then produces the rest of our basis vectors composing the Krylov subspace. This subspace is spanned by the eigenstates of the RPA matrix which has an overlap with the pivot vector. Thus, in general, the Arnoldi iterations can only be continued until this subspace is exhausted in which case the condition number goes to infinity. However, with finite numerical precision this maximum limit of Arnoldi iterations is further reduced.

In a typical iteration, during the first few iterations the condition number of the Krylov-space RPA matrix fluctuates, then approaches a stable plateau, and finally suddenly goes toward infinity. When that happens, the iteration must be stopped and one must backtrack to the iteration where the condition number was still acceptable. Therefore, the number of Arnoldi iterations can depend on the size of the HO basis, and the results presented in this section correspond to the numbers of iterations fixed according to this prescription.