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Embedded and non-embedded eigenvalues of a quantum graph for a carbon nanotube with a pentagon-based end cap
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DOI：10.1007/s40509-026-00391-0.png)
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Abstract
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In this paper, we consider a quantum graph corresponding to zigzag carbon nanotube Gamma 5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma <^>5$$\end{document} with 5-zigzags and a cap consisting of pentagons. It is well-known that the spectrum of Schr & ouml;dinger operator on zigzag carbon nanotube with 5-zigzags and without any cap consists of the infinitely many closed intervals (spectral bands) and infinitely many eigenvalues with infinite multiplicities (flat bands). On the other hand, we cap the zigzag carbon nanotube with 5-zigzags from one side. As a result, Schr & ouml;dinger operators with even potentials have additional eigenvalues. In this study, we notice that some cap-induced eigenvalues occur in spectral bands as embedded eigenvalues and some occur in spectral gaps. In this sense, we see that a cap plays the role of impurities for carbon nanotubes.
Keywords:
Carbon nanotube
Zigzag nanotube
Cap
Quantum graph
Eigenvalue
Spectral gap
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26
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222
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