the FWHM of the following approximate strength function: $$S(E)=-\frac{1}{\pi}{\rm Im}\left(
E-E_c-\sum_\alpha \frac{V_{c\alpha}^2}{E-\omega_\alpha+i\gamma_{22}/2}\right)^{-1},
\label{doorwaystr}$$ where, $E_c$ and $\omega_\alpha$ denote the energy of the GQR TD state and the energy of the $\alpha$th s.v.+1p1h state, respectively. $V_{c\alpha}$ represents the coupling matrix element between the GQR TD state and the s.v.+1p1h state $\alpha$.

Only $J^\pi=2^+$, $3^-$ modes are included as surface vibrations: We took only the lowest TD state as $J^\pi=2^+$ surface vibrational mode. On the other hand, we must pay attention to the collectivity of the octupole mode. Figure \[fig\_octstr\] shows the TDA strength function for the IS octupole operator. Compared with the experimental data [@spear], the energy of the lowest state is too high, and strengths are fragmented over several states. Thus, we took into account the lowest nine states for the octupole mode. Note that s.v.+1p1h states thus defined are not orthogonal. In this sense Eq. (\[doorwaystr\]) is an approximation which neglects the non-orthogonality.

![ TDA strength function for the IS octupole operator in $^{208}$Pb. []{data-label="fig_octstr"}](fig12.eps){width="6cm"}

The strength function based on Eq. (\[doorwaystr\]) is presented in Fig. \[fig\_doorwaystr\]. The width $\Gamma_2^{\rm (s.v.)}$ estimated by the FWHM is 0.074 MeV. This value is significantly smaller than the width $\Gamma_2=0.41$ MeV of 