\[sec:cal\_lsd\] for the normalization of the strengths. []{data-label="fig_strdis"}](fig3.eps){width="9.2cm"}

![ The $\Delta_3$ statistics for (a) $^{40}$Ca and (b) $^{208}$Pb. The horizontal axis $L$ shows the value of the energy interval for the unfolded spectrum. The solid curve represents the $\Delta_3$ for the GOE level fluctuation. See Fig. \[fig\_nns\] for other parameters. []{data-label="fig_delta3"}](fig4.eps){width="9.2cm"}

Results of local scaling dimension {#sec:results}
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Calculation of the local scaling dimension {#sec:cal_lsd}
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Since we are not interested in the global shape of the strength function, we actually adopt the normalized strength function $\bar{S}(E)$ for the fluctuation analysis as in the case of $^{40}$Ca [@aiba2]. The normalized strength function $\bar{S}(E)$ is given by $${\bar S}(E)=\sum_i{\bar S}_i\delta(E-{\bar E}_i+{\bar E}_0),
\label{eq_norstrfun}$$ where the normalized strength $\bar{S}_i $ of the $i$th level is defined by $$\bar{S}_i \equiv{\cal N}\frac{S_i\tilde{\rho}(E_i)}{
\tilde{S}(E_i)}.
\label{eq_norstr}$$ Here, $\tilde{\rho}(E)$ and $\tilde{S}(E)$ denote the level density and the strength function, respectively, smoothed by the Strutinsky method [@ring-schuck]. ${\cal N}$ is a normalization factor to guarantee $\sum_i\bar{S}_i=1$.

We determine the width parameter $\omega$ of the Strutinsky smoothing function as follows: We note that the smoothed strength function $\tilde{S}(E)$ should represent the global profile of the original strength function $S(E)$ at large energy scale, but at the same time, we 