with the fluctuation of strength function between $^{40}$Ca and $^{208}$Pb (Sec \[sec:comparison\]).

![ Schematic drawing of the doorway damping mechanism of the giant resonance, and related quantities. []{data-label="fig_pic_Ca"}](fig8.eps){width="7cm"}

Mechanism producing the total width {#sec:mechanism}
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### Landau damping

For $^{40}$Ca, the Landau damping is important, so that the strengths are already fragmented in the 1p1h levels. Therefore we first would like to investigate in $^{208}$Pb, how the strength is distributed in the TDA where only the 1p1h states are included.

![ TDA strength function for the IS quadrupole operator in $^{208}$Pb. See Fig. \[fig\_strfun\] for the dotted curve. []{data-label="fig_tdastr"}](fig9.eps){width="6cm"}

Figure \[fig\_tdastr\] shows the TDA strength function, which is obtained by means of the TDA, namely by neglecting 2p2h states, of the IS quadrupole operator. Different from the case of $^{40}$Ca, strengths in the GQR region is considerably concentrated on the single peak located at about 10.7 MeV. Because of this, the TDA strength function is very different from the full strength function in Fig. \[fig\_strfun\]. At the same time, we also see only a small effect of the Landau damping. In fact, the strength concentration on the single peak at $E=10.7$ MeV is 59% of the strengths in the energy 