A new approach for the analytic computation of the instantaneous normal modes spectrum

In the context of the instantaneous normal mode approach, the spectrum of t he Hessian of Hamiltonian is a key quantity to describe a liquids behaviour . The determination of the spectrum represents a major task for theoretical studies, and has recently been addressed in various works. In this work a new approach for the analytic computation of the Hessian spectrum is presen ted. The one-dimensional case for a system of particles interacting via a p urely repulsive potential at low density is analysed in detail and, also in the localized sector, the spectrum is computed exactly. Finally the possib le extensions of the method are discussed, together with a comparison with different approaches to the problem.