Fig. 4.

The vDOSs of the extended modes and the soft localized modes in the 3D model system. (A) Plot of the scaled vDOSs of the extended modes ($P^k>{\mathrm{\hspace{0.17em}10}}^{-2}$), ${\widehat{g}}_{\text{ex}}=g_{\text{ex}}(\omega )/A_0{\omega }_{\ast }^2$, against the scaled frequency $\widehat{\omega }=\omega /{\omega }_{\ast }$. (B) Plot of the vDOSs of the localized modes ($P^k<{\mathrm{\hspace{0.17em}10}}^{-2}$), $g_{\text{loc}}(\omega )$, against $\widehat{\omega }$. Insets are the same as the main images, but for the reduced vDOSs ${\widehat{g}}_{\text{ex}}/{\widehat{\omega }}^2$ in A and $g_{\text{loc}}/{\widehat{\omega }}^4$ in B. Below ${\omega }_{\text{ex0}}=\mathrm{\hspace{0.17em}0.066}{\omega }_{\ast }$, the vDOSs collapse onto $g_{\text{ex}}(\omega )=A_0{\omega }^2$ and $g_{\text{loc}}(\omega )={\alpha }_{\text{loc}}{(\omega /{\omega }_{\ast })}^4$ with ${\alpha }_{\text{loc}}=\mathrm{\hspace{0.17em}58}$.