Eigenvalue accumulation for operator convolutions on locally compact groups

Within the framework of quantum harmonic analysis, for a locally compact group $G$ with a square-integrable, irreducible unitary representation, we analyze the eigenvalue distributions of convolutions between indicator functions on $G$ and a fixed density operator on the representation space, a concept which generalizes localization operators. In particular, we consider a sequence of such operators and the asymptotic number of eigenvalues that lie within a small distance of $1$. We show that a previously postulated type of asymptotic behavior occurs if and only if the group is unimodular and the sets underlying the indicator functions form a Følner sequence. Applying this, we obtain positive results for nilpotent and homogeneous Lie groups, recovering an established result for the Heisenberg group as a special case.