In Section 5.1, we showed our strength functions calculated
with 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
shells, about twenty more
iterations were needed, compared to calculations made with
shells. In Figs. 7, 8, and 9, we show
the convergence of strength functions for the
,
, and
modes, respectively. Each panel shows the difference of two
strength functions obtained in the intervals of
HO
shells, between
of 22 and 38.
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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 strength functions calculated for
, 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.
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As noted in Ref. [5], well before the maximum number of
iterations (equal to the RPA dimension ) 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.