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 compu te the exact spectrum also in the localized sector. We finally perform a nu merical analysis of the localization properties of the eigenfunctions.