标题： 边缘变量下仿射谱编码中的体Weyl渐近性 显示英文标题

标题： Bulk Weyl Asymptotics in the Edge Variable under Affine Spectral Encoding

摘要： 我们证明了一个Tauberian传递原理，表明对于任何紧致闭合的黎曼流形$(M^d,g)$，仿射谱编码$C=\pi-\epsilon\lambda$将拉普拉斯Weyl渐近性传输到主体区域$C\to-\infty$中的边变量的Weyl定律（等价地，$(\pi-C)/\epsilon\to\infty$）：$N_{\mu_C}(C)\sim \gamma_d \epsilon^{-d/2}(\pi-C)^{d/2}$和$\rho_{\mathrm{bulk}}(C)\sim \frac{d}{2}\gamma_d \epsilon^{-d/2}(\pi-C)^{(d-2)/2}$，因此$d$和Weyl常数$\gamma_d$可以从一维边变量数据中恢复。 相反，整体幂律 $N_{\mu_C}(C)\sim A(\pi-C)^\alpha$ 作为 $C\to-\infty$ 意味着 $d=2\alpha$ 和 $\gamma_d=A\epsilon^{d/2}$。 我们证明了在多项式类型编码$g(\lambda)=a-b\lambda^{k}L(\lambda)$（边变量指数强制$k=1$）和小扰动下的稳定性$C=\pi-\epsilon\lambda+\delta(\lambda)$，以及$\delta(\lambda)=o(\lambda)$的唯一性。对于常曲率模型空间，我们记录了热迹和谱ζ函数的加强对应关系，$H_{\mathrm{edge}}(s)=\Theta_\Delta(\epsilon s)$和$\zeta_{\mathrm{edge}}(u)=\epsilon^{-u}\zeta_\Delta(u)$，并通过广义的一维模型（Krein弦）实现了重数。 当存在一个Weyl余项$O(\Lambda^{(d-1)/2})$时，它会转移到变量为$C$的整体余项$O((\pi-C)^{(d-1)/2})$。 显示英文摘要

摘要： We prove a Tauberian transfer principle showing that for any compact closed Riemannian manifold $(M^d,g)$ the affine spectral encoding $C=\pi-\epsilon\lambda$ transports Laplacian Weyl asymptotics to a Weyl law in the edge variable in the bulk regime $C\to-\infty$ (equivalently, $(\pi-C)/\epsilon\to\infty$): $N_{\mu_C}(C)\sim \gamma_d \epsilon^{-d/2}(\pi-C)^{d/2}$ and $\rho_{\mathrm{bulk}}(C)\sim \frac{d}{2}\gamma_d \epsilon^{-d/2}(\pi-C)^{(d-2)/2}$, so that $d$ and the Weyl constant $\gamma_d$ are recoverable from one-dimensional edge-variable data. Conversely, a bulk power law $N_{\mu_C}(C)\sim A(\pi-C)^\alpha$ as $C\to-\infty$ implies $d=2\alpha$ and $\gamma_d=A\epsilon^{d/2}$. We establish the uniqueness of the affine rule among polynomial-type encodings $g(\lambda)=a-b\lambda^{k}L(\lambda)$ (the edge-variable exponent forces $k=1$) and stability under small perturbations $C=\pi-\epsilon\lambda+\delta(\lambda)$ with $\delta(\lambda)=o(\lambda)$. For constant-curvature model spaces we record strengthened correspondences for heat traces and spectral zeta, $H_{\mathrm{edge}}(s)=\Theta_\Delta(\epsilon s)$ and $\zeta_{\mathrm{edge}}(u)=\epsilon^{-u}\zeta_\Delta(u)$, and we realize multiplicities via generalized one-dimensional models (Krein strings). When a Weyl remainder $O(\Lambda^{(d-1)/2})$ is available, it transfers to a bulk remainder $O((\pi-C)^{(d-1)/2})$ in the $C$-variable.