Analytic Computation of the Instantaneous Normal Modes Spectrum in Low-Density Liquids

We analytically compute the spectrum of the Hessian of the Hamiltonian for a system of N particles interacting via a purely repulsive potential in one dimension. Our approach is valid in the low-density regime, where we compute the exact spectrum also in the localized sector. We finally perform a numerical analysis of the localization properties of the eigenfunctions.