Application of the random matrix theory to the boson peak in glasses

The density of vibrational states $g(\omega )$ of an amorphous system is studied by using the random-matrix theory. Taking into account the most important correlations between elements of the random matrix of the system, equations for the density of vibrational states $g(\omega )$ are obtained. The analysis of these equations shows that in the low-frequency region the vibrational density of states has the Debye behavior $g(\omega )\sim {\omega }^2$. In the higher frequency region, there is the boson peak as an additional contribution to the density of states. The obtained equations are in a good agreement with the numerical results and allow us to find an exact shape of the boson peak.