Scattering by a medium with self-similar or fractal structure

Abstract

We develop a topological framework for wave scattering by a medium with self-similar or prefractal geometry. Physical scaling identities motivate an auxiliary boundary-integral model that is periodic in logarithmic scale, and the 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 subspace of the full boundary symbol. Under the stated spectral-isolation and point-gap conditions, we prove that the determinant winding of the exact Riesz-reduced family agrees with that of the projected matrix. For the regular-simplex configurations, we obtain nonzero winding formulas and track the resulting local winding data across finite prefractal levels and along a geometric sequence of wavenumbers. We also obtain conditional results for different dilation centers and show that the Zak phase of the chiral Hermitianization is equal to $π$ times the point-gap winding modulo $2π$. Thus, the scale-periodic boundary model admits a rigorous topological reduction.

Disclosure

“r phases: they do not determine a bandwise Zak phase without a globally isolated simple band. They do, however, determine the Zak phase of the negative spectral subspace of the associated chiral Hermitianization. Disclaimer. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process. During the preparation of this work, the authors used ChatGPT to assist with language editing and code writing for numerical experiments. After using this tool/service, th”

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Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-23T05:12:30.404844Z

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