具有自相似或分形结构介质的散射
Topological properties and a multiplicative Bloch-Floquet-Zak transform for scattering by self-similar or fractal media
AI总结：
该研究构建了自相似或分形介质波散射的拓扑框架，通过边界积分模型与Riesz投影实现严格拓扑约化，得到正则单形构型的非零环绕数公式等结果。
AI中文摘要：
我们开发了用于具有自相似或预分形几何介质的波散射的拓扑框架。物理标度恒等式催生了对数尺度上周期的辅助边界积分模型，Bloch-Floquet-Zak变换将其跨尺度耦合纤维化。对于足够小且间距良好的分量，平衡密度定义了可计算的有限维投影矩阵，而Riesz投影则选取完整边界符号对应的不变谱子空间。在给定的谱隔离和点间隙条件下，我们证明精确Riesz约化族的行列式环绕数与投影矩阵的一致。对于正则单形构型，我们得到非零环绕数公式，并追踪有限预分形层级及波数几何序列上的局部环绕数数据。我们还针对不同膨胀中心得到了条件结果，且手性厄米化的Zak相位等于π乘以点间隙环绕数模2π。因此，该标度周期边界模型容许严格的拓扑约化。
英文摘要：
We develop the first rigorous topological framework for an auxiliary boundary-integral model motivated by physical scaling identities for wave scattering from self-similar or fractal media. The model is periodic in logarithmic scale, and a multiplicative Bloch-Floquet-Zak transform fiberizes its interscale coupling. For sufficiently small, well-separated components, equilibrium densities define a computable finite-dimensional projected matrix, while Riesz projections select the corresponding invariant spectral subspaces of the full boundary symbol. Under suitable spectral-isolation and point-gap conditions, and after recentering the two families at their respective same-scale reference values, we prove that the determinant winding of the exact Riesz-reduced family agrees with that of the projected matrix family. For regular-simplex configurations, we derive explicit nonzero winding formulas and track the resulting local winding data across finite prefractal levels and along a geometric sequence of wavenumbers. We also formulate conditional winding data for multiple dilation centers and show that the Zak phase of the chiral Hermitianization equals $π$ times the point-gap winding modulo $2π$. Thus, the scale-periodic boundary model admits a rigorous topological reduction.
