straightforward way, we can define the local scaling dimension as, $$D_m(\epsilon)\equiv \frac{1}{m-1}
\frac{\partial\log\chi_m(\epsilon)}
{\partial\log\epsilon}.
\label{scaledim}$$ Since the local scaling dimension has a definite physical meaning similar to that of the generalized fractal dimension, the value of $D_m(\epsilon)$ can quantitatively characterize the fluctuation of the strength function at each energy scale $\epsilon$.

In the actual calculation of the local scaling dimension, we define it by means of the finite difference under the change of a factor 2, $$D_m(\sqrt{2}\epsilon) = \frac{1}{m-1}\frac{
\log\chi_m(2\epsilon)-
\log\chi_m(\epsilon)}
{\log 2},
\label{approscaledim}$$ rather than the derivative in Eq. (\[scaledim\]).

Numerical Calculation of Strength Function {#sec:numerical}
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We calculated the strength function of the IS GQR in $^{208}$Pb within the second TDA including the 1p1h and 2p2h excitations. Single-particle wave-functions and energies were obtained for a Woods-Saxon potential including the Coulomb interaction. The effective mass parameter $m^*/m$, which scales the Woods-Saxon single-particle energies $\varepsilon_{\rm WS}$ as $\varepsilon_{\rm HF}=\varepsilon_{\rm WS}/(m^*/m)$ to simulate the bare (Hartree-Fock) single-particle energies $\varepsilon_{\rm HF}$, is set to be 1 in this calculation.

As the residual interaction, the Landau-Migdal-type interaction [@Schwe] including the density-dependence was adopted. The model space was constructed in terms of single-particle states within the four major shells, two below and two above 