we can not find the difference between $^{208}$Pb and $^{40}$Ca. Studies of fluctuation at intermediate energy scales lead to the finding of the difference. In the following we shall investigate the mechanism which brings about the difference in fluctuations at intermediate energy scales.

Studies of damping process {#sec:damping}
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Let us now investigate origins of the difference between the cases of $^{40}$Ca and $^{208}$Pb. In our previous study of the GQR in $^{40}$Ca, we have shown that the behavior of the local scaling dimension, shown in Fig. \[fig\_pat\_ldim\] (d), can be interpreted in terms of the doorway damping mechanism. We here employ the same picture in order to clarify the damping mechanism of the GQR in $^{208}$Pb.

The doorway damping mechanism consists of a two-step process which is illustrated in Fig. \[fig\_pic\_Ca\]. The giant resonance is spread over the 1p1h states due to the Landau damping, the width of which is denoted by $\Gamma_{\rm L}$. The average spacing of 1p1h states is denoted by $D_{\rm 1p1h}$. The 1p1h states are considered here as the “doorway" states of the damping process. The 1p1h states then couple to more complicated background states (2p2h states) through the residual two-body interaction. The coupling causes 