Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws

TL;DR

This paper demonstrates that certain operators with superlinear eigenvalue growth cannot be approximated by single-valuation exponential kernels, revealing a fundamental spectral dimension obstruction.

Contribution

It establishes a spectral obstruction showing that operators with superlinear eigenvalue counting cannot be limits of single-valuation kernels, highlighting a key structural difference.

Findings

Single-valuation exponential kernels converge to a fourth-order operator with spectral dimension 1/2.

Operators with superlinear eigenvalue growth have spectral dimension 2.

Spectral dimension invariance implies these classes of operators are fundamentally incompatible.

Abstract

We show that single-valuation exponential kernels, under mild regularity assumptions, converge in the continuum limit to a fourth-order operator with heat asymptotics $\Theta(t)\sim t^{-1/4}$ and hence spectral dimension $d_s=\tfrac12$. Independently, a Tauberian analysis implies that any self-adjoint operator with superlinear eigenvalue counting $N(\lambda)\sim \lambda\,L(\lambda)$ must satisfy $\Theta(t)\sim t^{-1}L(1/t)$ 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.