Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws

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.