Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals Article Swipe

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Matthew J. Colbrook , Mark Embree , Jake Fillman ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2407.20353

We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and other spectral characteristics. Our motivation comes from the study of almost-periodic operators, particularly those that arise as models of quasicrystals. Such operators are known for intricate hierarchical patterns and often display delicate spectral properties, such as Cantor spectra, which are significant in studying quantum mechanical systems and materials science. We propose a series of algorithms that compute these properties under different assumptions and explore their theoretical implications through the Solvability Complexity Index (SCI) hierarchy. This approach provides a rigorous framework for understanding the computational feasibility of these problems, proving algorithmic optimality, and enhancing the precision of spectral analysis in practical settings. For example, we show that our methods are optimal by proving certain lower bounds (impossibility results) for the class of limit-periodic Schrödinger operators. We demonstrate our methods through state-of-the-art computations for aperiodic systems in one and two dimensions, effectively capturing these complex spectral characteristics. The results contribute significantly to connecting theoretical and computational aspects of spectral theory, offering insights that bridge the gap between abstract mathematical concepts and their practical applications in physical sciences and engineering. Based on our work, we conclude with conjectures and open problems regarding the spectral properties of specific models.

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Quasicrystal Algorithm Computer science Mathematics Geometry

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2407.20353
PDF: https://arxiv.org/pdf/2407.20353
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4401202788

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DOI: https://doi.org/10.48550/arxiv.2407.20353

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Title: Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals

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Type: preprint

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Language: en

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Publication year: 2024

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Publication date: 2024-07-29

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Authors: Matthew J. Colbrook, Mark Embree, Jake Fillman

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Landing page: https://arxiv.org/abs/2407.20353

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PDF URL: https://arxiv.org/pdf/2407.20353

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OA URL: https://arxiv.org/pdf/2407.20353

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Concepts: Quasicrystal, Algorithm, Computer science, Mathematics, Geometry

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Cited by: 0

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Related works (count): 10

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