Eigenvalue accumulation for operator convolutions on locally compact groups

TL;DR

This paper investigates the eigenvalue distribution of convolutions in quantum harmonic analysis on locally compact groups, revealing asymptotic behaviors linked to group properties like unimodularity and F{46}lner sequences.

Contribution

It establishes conditions under which eigenvalue asymptotics occur, generalizing localization operators to a broad class of groups and recovering known results for specific cases.

Findings

Eigenvalue distribution follows specific asymptotics for unimodular groups.

Asymptotic behavior occurs iff the group is unimodular and the sets form a F{46}lner sequence.

Results apply to nilpotent and homogeneous Lie groups, including the Heisenberg group.

Abstract

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 F{\o}lner 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.