AI 中文总结
研究离散猫映射在有限环面上算子的对角格林函数，证明其精确公式，揭示对角景观是轨道结构的谱不变量，还建立相关恒等式，表明拉普拉斯扰动对定位间隙的影响，结果经计算验证。
AI 中文摘要
我们研究了有限环面\((\mathbb{Z}/N\mathbb{Z})^2\)上算子\(L_N = I - \alpha P\)的对角格林函数\(\widetilde{u}(x)=[L_N^{-1}]_{x,x}\)，其中\(P\)是离散猫映射\(T_N(x)=Ax \bmod N\)的转移算子。我们证明了精确公式\(\widetilde{u}(x)=(1 - \alpha^{k_x})^{-1}\)，其中\(k_x\)是\(x\)在\(T_N\)下的最小周期。该公式显示对角景观是轨道结构的完整谱不变量，仅通过轨道长度依赖于每个点。由于\(\det(A - I)= -1\)是\(\mathbb{Z}/N\mathbb{Z}\)中对每个\(N\geq2\)的单位，原点是\(T_N\)的唯一不动点和\(\widetilde{u}\)的唯一全局最大值。这种定位仅由算术驱动，无无序且无对称性破缺，是一种不同于经典安德森理论和菲洛切 - 马约罗达景观理论的机制。我们还建立了钱德拉格林 - 泽塔恒等式，表明格林迹满足\(\operatorname{tr}(G_N)=N^2 - \alpha\frac{d}{d\alpha}\log Z_N(\alpha)\)，其中\(Z_N\)是\(T_N\)的动态泽塔函数，并且拉普拉斯扰动在\(\varepsilon\)的一阶上降低了定位间隙。所有结果均通过计算验证。
英文摘要
We study the diagonal Green function $\widetilde{u}(x)=[L_N^{-1}]_{x,x}$ of the operator $L_N=I-αP$ on the finite torus $(\mathbb{Z}/N\mathbb{Z})^2$, where $P$ is the transfer operator of the discrete cat map $T_N(x)=Ax \bmod N$. We prove the exact formula $\widetilde{u}(x)=(1-α^{k_x})^{-1}$, where $k_x$ is the minimal period of $x$ under $T_N$. This formula appears to be new. It shows that the diagonal landscape is a complete spectral invariant of the orbit structure, depending on each point only through its orbit length. Since $\det(A-I)=-1$ is a unit in $\mathbb{Z}/N\mathbb{Z}$ for every $N\ge2$, the origin is the unique fixed point of $T_N$ and the unique global maximum of $\widetilde{u}$. The resulting localization is driven by arithmetic alone, with no disorder and no broken symmetry, a mechanism distinct from classical Anderson theory and from Filoche--Mayboroda landscape theory. We further establish the Chandra Green--Zeta Identity, showing that the Green trace satisfies $\operatorname{tr}(G_N)=N^2-α\frac{d}{dα}\log Z_N(α)$, where $Z_N$ is the dynamical zeta function of $T_N$, and that a Laplacian perturbation degrades the localization gap at first order in $\varepsilon$. All results are verified computationally.
Comments15 pages, 6 figures
