{"title":"Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals","authors":"Matthew J. Colbrook, Mark Embree, Jake Fillman","doi":"arxiv-2407.20353","DOIUrl":null,"url":null,"abstract":"We introduce computational strategies for measuring the ``size'' of the\nspectrum of bounded self-adjoint operators using various metrics such as the\nLebesgue measure, fractal dimensions, the number of connected components (or\ngaps), and other spectral characteristics. Our motivation comes from the study\nof almost-periodic operators, particularly those that arise as models of\nquasicrystals. Such operators are known for intricate hierarchical patterns and\noften display delicate spectral properties, such as Cantor spectra, which are\nsignificant in studying quantum mechanical systems and materials science. We\npropose a series of algorithms that compute these properties under different\nassumptions and explore their theoretical implications through the Solvability\nComplexity Index (SCI) hierarchy. This approach provides a rigorous framework\nfor understanding the computational feasibility of these problems, proving\nalgorithmic optimality, and enhancing the precision of spectral analysis in\npractical settings. For example, we show that our methods are optimal by\nproving certain lower bounds (impossibility results) for the class of\nlimit-periodic Schr\\\"odinger operators. We demonstrate our methods through\nstate-of-the-art computations for aperiodic systems in one and two dimensions,\neffectively capturing these complex spectral characteristics. The results\ncontribute significantly to connecting theoretical and computational aspects of\nspectral theory, offering insights that bridge the gap between abstract\nmathematical concepts and their practical applications in physical sciences and\nengineering. Based on our work, we conclude with conjectures and open problems\nregarding the spectral properties of specific models.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"173 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Spectral Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.20353","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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 Schr\"odinger 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.