  The vibrational anomalies of glasses, in particular the boson peak, are
addressed from the standpoint of heterogeneous elasticity, namely the spatial
fluctuations of elastic constants caused by the structural disorder of the
amorphous materials. In the first part of this review article a mathematical
analogy between diffusive motion in a disordered environment and a scalar
simplification of vibrational motion under the same condition is emploited. We
demonstrate that the disorder-induced long-time tails of diffusion correspond
to the Rayleigh scattering law in the vibrational system and that the
cross-over from normal to anomalous diffusion corresponds to the boson peak.
The anomalous motion arises as soon as the disorder-induced self-energy exceeds
the frequency-independent diffusivity/elasticity. For this model a variational
scheme is emploited for deriving two mean-field theories of disorder, the
self-consistent Born approximation (SCBA) and coherent-potential approximation
(CPA). The former applies if the fluctuations are weak and Gaussian, the latter
applies for stronger and non-Gaussian fluctuations. In the second part the
vectorial theory of heterogenous elasticity is presented and solved in SCBA and
CPA, introduced for the scalar model. Both approaches predict and explain the
boson-peak and the associated anomalies, namely a dip in the acoustic phase
velocity and a characteristic strong increase of the acoustic attenuation below
the boson peak. Explicit expressions for the density of states and the
inelastic Raman, neutron and X-ray scattering laws are given. Recent
conflicting ways of explaining the boson-peak anomalies are discussed.
