Bulk Weyl Asymptotics in the Edge Variable under Affine Spectral Encoding

Abstract

We prove a Tauberian transfer principle showing that for any compact closed Riemannian manifold $(M^d,g)$ the affine spectral encoding $C=π-ελ$ transports Laplacian Weyl asymptotics to a Weyl law in the edge variable in the bulk regime $C\to-\infty$ (equivalently, $(π-C)/ε\to\infty$): $N_{μ_C}(C)\sim γ_d ε^{-d/2}(π-C)^{d/2}$ and $ρ_{\mathrm{bulk}}(C)\sim \frac{d}{2}γ_d ε^{-d/2}(π-C)^{(d-2)/2}$, so that $d$ and the Weyl constant $γ_d$ are recoverable from one-dimensional edge-variable data. Conversely, a bulk power law $N_{μ_C}(C)\sim A(π-C)^α$ as $C\to-\infty$ implies $d=2α$ and $γ_d=Aε^{d/2}$. We establish the uniqueness of the affine rule among polynomial-type encodings $g(λ)=a-bλ^{k}L(λ)$ (the edge-variable exponent forces $k=1$) and stability under small perturbations $C=π-ελ+δ(λ)$ with $δ(λ)=o(λ)$. For constant-curvature model spaces we record strengthened correspondences for heat traces and spectral zeta, $H_{\mathrm{edge}}(s)=Θ_Δ(εs)$ and $ζ_{\mathrm{edge}}(u)=ε^{-u}ζ_Δ(u)$, and we realize multiplicities via generalized one-dimensional models (Krein strings). When a Weyl remainder $O(Λ^{(d-1)/2})$ is available, it transfers to a bulk remainder $O((π-C)^{(d-1)/2})$ in the $C$-variable.

Theorems & Definitions (94)

• theorem thmcountertheorem: Main Theorem
• remark thmcounterremark: Terminology
• definition thmcounterdefinition: Spectral measure and counting function
• lemma thmcounterlemma: Counting function properties
• proof
• lemma thmcounterlemma: Monotonicity of the encoding
• proof
• remark thmcounterremark
• lemma thmcounterlemma: Exact composition identity
• proof