  We discuss vibrational localization problems in glasses and disordered media
in this paper. It is claimed that the essence of the localization problem is
already observed in disordered lattice models. These kinds of vibrations belong
to a different universality class than bonded electrons. Specifically, The
eigenvectors are extended even in two dimensions. Moreover, the correlation
exponent does not diverge in the transition from localized to delocalized
states. Furthermore, the volume of the extended states is scaled according to
the distance from the transition as $V\sim |\omega-\omega_c| L^d$.
Interestingly, boson peaks can be observed in the density of states in both two
and three dimensions. We studied the eigenstates of this problem and analyzed
the scattering effects in these lattices. Importantly, we found that in two
dimensions the boson peak, the localization edge, and the beginning of
anomalous scattering are at the same frequency. In three dimensions, however,
there are three separate regions: (1) localized, (2) weakly scattered, and (3)
anomalously scattered. Finally, we discuss the relevance of this study to
actual experiments and glasses.
