Eigenvalue accumulation for operator convolutions on locally compact groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915846063915008 |
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| author | Schroth, Florian |
| author_facet | Schroth, Florian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08141 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvalue accumulation for operator convolutions on locally compact groups Schroth, Florian Functional Analysis Mathematical Physics 47B90 (Primary) 43A30, 43A65, 43A07, 47A75 (Secondary) 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. |
| title | Eigenvalue accumulation for operator convolutions on locally compact groups |
| topic | Functional Analysis Mathematical Physics 47B90 (Primary) 43A30, 43A65, 43A07, 47A75 (Secondary) |
| url | https://arxiv.org/abs/2603.08141 |