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

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