加倍度量测度空间中有限相互作用范围算子的谱与伪谱近似
Spectral and Pseudospectral Approximation of Finite-Interaction-Range Operators in Doubling Metric Measure Spaces
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中文总结 AI 辅助
该研究针对加倍度量测度空间上的有限相互作用范围算子，通过交换子估计得到双侧伪谱包含界与Hausdorff收敛结果，覆盖多种几何场景并给出误差可控的谱近似方案。
中文摘要 AI 辅助
我们研究左不变加倍度量测度空间$(X,d,\mu)$的一致离散子集$\Gamma$上$\ell^2(\Gamma)$中有界有限相互作用范围算子$H$，目标是从支撑在球$B_L(x)\cap\Gamma$上的有限截面$H_{L,x}$近似$H$的谱信息。主要技术输入是Lipschitz“帐篷”局部化子$W_{L,x}$的交换子估计，其仅依赖几何性质（加倍性）和一致相互作用度。由此得到显式双侧伪谱包含界：$\gamma_{L,\varepsilon}(H)\subset \sigma_\varepsilon(H)\subset \gamma_{L,\varepsilon+C_0/L}(H)$，且当$L\to\infty$时窗口伪谱的Hausdorff收敛到（全局）伪谱。在自伴/正规情形下，这产生可计算的间隙检验和带严格$O(1/L)$误差控制的谱采样方案；在非正规情形下，得到伪谱的对应近似结果。该框架分离了此前$\mathbb{R}^n$和可数阿贝尔群方法的几何核心，覆盖准晶模型的不规则几何，以及离散（非阿贝尔）幂零群等新情形（例如离散海森堡群）。
英文摘要
We study bounded finite-interaction-range operators $H$ on $\ell^2(Γ)$, where $Γ$ is a uniformly discrete subset of a left-invariant doubling metric measure space $(X,d,μ)$. Our goal is to approximate spectral information of $H$ from finite sections $H_{L,x}$ supported on balls $B_L(x)\capΓ$. The main technical input is a commutator estimate for Lipschitz ''tent'' localisations $W_{L,x}$, which depends only on geometric properties (doubling) and a uniform interaction degree. As a consequence, we obtain explicit two-sided pseudospectral inclusion bounds of the form \[ γ_{L,\varepsilon}(H)\subset σ_\varepsilon(H)\subset γ_{L,\varepsilon+C_0/L}(H), \] and Hausdorff convergence of window pseudospectra to the (global) pseudospectrum as $L\to\infty$. In the self-adjoint/ normal case this yields computable gap tests and spectral sampling schemes with rigorous $O(1/L)$ error control, while in the non-normal case it leads to corresponding approximation results for pseudospectra. The framework isolates the geometric core behind earlier approaches on $\mathbb{R}^n$ and on countable Abelian groups, and covers irregular geometries arising in quasicrystal models, as well as new cases such as discrete (non-Abelian) nilpotent groups (including for example the discrete Heisenberg group).
