Spectral continuity for aperiodic quantum systems
- This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson-Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay-Rudin-Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds theThis work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson-Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay-Rudin-Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.…
Author details: | Siegfried BeckusORCiDGND, Jean BellissardORCiDGND, Giuseppe De NittisORCiDGND |
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DOI: | https://doi.org/10.1063/5.0011488 |
ISSN: | 0022-2488 |
ISSN: | 1089-7658 |
Title of parent work (English): | Journal of mathematical physics |
Subtitle (English): | applications of a folklore theorem |
Publisher: | American Institute of Physics |
Place of publishing: | Melville, NY |
Publication type: | Article |
Language: | English |
Date of first publication: | 2020/12/23 |
Publication year: | 2020 |
Release date: | 2023/07/13 |
Volume: | 61 |
Issue: | 12 |
Article number: | 123505 |
Number of pages: | 19 |
Funding institution: | Mathematics Department at Technion, Israel; Facultad de Matematicas at; the Pontificia Universidad Catolica, Chile; Department of Mathematics; Westfalische Wilhelms-Universitat, Munster, Germany; Georgia Institute; of Technology, USA; Erwin Schrodinger Institute, Vienna; Research; Training Group at the Friedrich-Schiller University in Jena, Germany; [1523/2]; Mathematisches Forschungsinstitut Oberwolfach; National; Science FoundationNational Science Foundation (NSF) [DMS1160962]; FONDECYTComision Nacional de Investigacion Cientifica y Tecnologica; (CONICYT)CONICYT FONDECYT [1190204] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |