  We study the statistical properties of eigenvalues of the Hessian matrix
${\cal H}$ (matrix of second derivatives of the potential energy) for a
classical atomic liquid, and compare these properties with predictions for
random matrix models (RMM). The eigenvalue spectra (the Instantaneous Normal
Mode or INM spectra) are evaluated numerically for configurations generated by
molecular dynamics simulations. We find that distribution of spacings between
nearest neighbor eigenvalues, s, obeys quite well the Wigner prediction $s
exp(-s^2)$, with the agreement being better for higher densities at fixed
temperature. The deviations display a correlation with the number of localized
eigenstates (normal modes) in the liquid; there are fewer localized states at
higher densities which we quantify by calculating the participation ratios of
the normal modes. We confirm this observation by calculating the spacing
distribution for parts of the INM spectra with high participation ratios,
obtaining greater conformity with the Wigner form. We also calculate the
spectral rigidity and find a substantial dependence on the density of the
liquid.
