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We study the spectrum of a system of coupled disordered harmonic oscillatorsin the thermodynamic limit. This Euclidean random matrix ensemble has beensuggested as model for the low-temperature vibrational properties of glass.Exact numerical diagonalization is performed in three and two spatialdimensions, which is accompanied by a detailed finite size analysis. It revealsa low-frequency regime of sound waves that are damped by Rayleigh scattering.At large frequencies localized modes exist. In between, the central peak in thevibrational density of states is well described by Wigner's semicircle law fornot too large disorder, as is expected for simple random matrix systems. Wecompare our results with predictions from two recent self-consistent fieldtheories.

Properties of stable ensembles of Euclidean random matrices