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We prove uniqueness of self-similar profiles for the one-dimensionalinelastic Boltzmann equation with moderately hard potentials, that is withcollision kernel of the form | $\bullet$ | $\gamma$ for $\gamma$ > 0 smallenough (explicitly quantified). Our result provides the first uniquenessstatement for self-similar profiles of inelastic Boltzmann models allowing forstrong inelasticity besides the explicitly solvable case of Maxwellinteractions (corresponding to $\gamma$ = 0). Our approach relies on aperturbation argument from the corresponding Maxwell model through a carefulstudy of the associated linearised operator. In particular, a part of the paperis devoted to the trend to equilibrium for the Maxwell model in suitableweighted Sobolev spaces, an extension of results which are known to hold inweaker topologies. Our results can be seen as a first step towards a fullproof, in the one-dimensional setting, of a conjecture in Ernst \& Brito (2002)regarding the determination of the long-time behaviour of solutions toinelastic Boltzmann equation.

One-dimensional inelastic Boltzmann equation: Regularity \& uniqueness of self-similar profiles for moderately hard potentials