Analytic computation of the Instantaneous Normal Modes spectrum in low density liquids

TL;DR

This paper analytically derives the spectrum of the Hessian matrix for low-density one-dimensional liquids, providing exact results in the localized sector and analyzing eigenfunction localization.

Contribution

It introduces an analytical method to compute the Hessian spectrum in low-density liquids, including localized eigenstates, which was not previously achieved.

Findings

Exact Hessian spectrum in low-density regime

Localization properties of eigenfunctions analyzed

Numerical results support analytical findings

Abstract

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.