For a schematic shell model, we show numerically that, contrary to the behaviour of eigenfunctions, the shapes ofthe so-called local spectral density of states become close to their forms at extremely strong perturbation (afterrescaling) even when the perturbation is relatively weak. The same phenomenon is also found for the randomversion of the schematic shell model. We suggest that this property of the local spectral density of states may becommon to models in which the Hamiltonian matrices in independent particle states have a banded and regularstructure.