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 $\hat{I}˜(t)\sim t^{-1/4}$ and hence spectral dimension $d_s=\frac{1}{2}$. Independently, a Tauberian analysis implies that any self-adjoint operator with superlinear eigenvalue counting $N(\hat{I}»)\sim \hat{I}»L(\hat{I}»)$ must satisfy $\hat{I}˜(t)\sim t^{-1}L(1/t)$ 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.

31 pages