Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws
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| Format: | Preprint |
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2026
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| author | Watson, Douglas F. Valentinuzzi, Tiziano |
| author_facet | Watson, Douglas F. Valentinuzzi, Tiziano |
| contents | We show that single-valuation exponential kernels, under mild regularity assumptions, converge in the continuum limit to a fourth-order operator with heat asymptotics $Θ(t)\sim t^{-1/4}$ and hence spectral dimension $d_s=\tfrac12$. Independently, a Tauberian analysis implies that any self-adjoint operator with superlinear eigenvalue counting $N(λ)\sim λ\,L(λ)$ must satisfy $Θ(t)\sim t^{-1}L(1/t)$ and therefore has spectral dimension $d_s=2$. Since spectral dimension is invariant under unitary equivalence and compact perturbations, these exponents are incompatible, yielding a structural obstruction that separates single-valuation kernel limits from operators with accelerated spectral growth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_00052 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws Watson, Douglas F. Valentinuzzi, Tiziano Spectral Theory Number Theory 35P20 (Primary) 58J35, 11M26, 47A10 (Secondary) We show that single-valuation exponential kernels, under mild regularity assumptions, converge in the continuum limit to a fourth-order operator with heat asymptotics $Θ(t)\sim t^{-1/4}$ and hence spectral dimension $d_s=\tfrac12$. Independently, a Tauberian analysis implies that any self-adjoint operator with superlinear eigenvalue counting $N(λ)\sim λ\,L(λ)$ must satisfy $Θ(t)\sim t^{-1}L(1/t)$ and therefore has spectral dimension $d_s=2$. Since spectral dimension is invariant under unitary equivalence and compact perturbations, these exponents are incompatible, yielding a structural obstruction that separates single-valuation kernel limits from operators with accelerated spectral growth. |
| title | Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws |
| topic | Spectral Theory Number Theory 35P20 (Primary) 58J35, 11M26, 47A10 (Secondary) |
| url | https://arxiv.org/abs/2604.00052 |