In the context of the Instantaneous Normal Mode approach, the spectrum of the Hessian of Hamiltonian is a key quantity to describe liquids behaviour. The determination of the spectrum represents a major task for theoretical studies, and has been addressed recently in various works. In this paper a new approach for the analytic computation of the Hessian spectrum is presented. The one dimensional case for a system of particles interacting via a purely repulsive potential at low density is analyzed in details and the spectrum is computed exactly also in the localized sector. Finally, the possible extensions of the method are discussed, together with a comparison with different approaches to the problem.