Niikuni, Hiroaki On the spectra of Schrödinger operators on zigzag nanotubes with multiple bonds. (English) Zbl 1423.81084 Funkc. Ekvacioj, Ser. Int. 62, No. 2, 255-283 (2019). Summary: In this paper, we study the spectral structure of periodic Schrödinger operators on a generalization of carbon nanotubes from the point of view of the quantum graphs. Since there exist chemical double bonds between carbon atoms on a hexagonal lattice with a cylindrical structure corresponding to carbon nanotubes, we study the spectral structure of periodic Schrödinger operators on zigzag nanotubes with multiple bonds of atoms in this paper. Utilizing the Floquet-Bloch theory, the spectrum of the operator consists of the absolutely continuous spectral bands and the flat band. We study the relationship between the number of the chemical bonds and the existence of spectral gaps. Cited in 1 Document MSC: 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices 34L05 General spectral theory of ordinary differential operators 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 34B45 Boundary value problems on graphs and networks for ordinary differential equations 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 82D77 Quantum waveguides, quantum wires 82D80 Statistical mechanics of nanostructures and nanoparticles Keywords:carbon nanotube; zigzag nanotube; quantum graph; spectral gap; band structure; Floquet-Bloch theory; Hill operator PDFBibTeX XMLCite \textit{H. Niikuni}, Funkc. Ekvacioj, Ser. Int. 62, No. 2, 255--283 (2019; Zbl 1423.81084) Full Text: DOI References: [1] Brown, B. M., Eastham, M. S. P. and Schmidt, K. M., Periodic Differential Operators. Operator Theory: Advances and Applications, Birkhäuser, Basel (2013). [2] Brüning, J., Geyler, V. and Pankrashkin, K., Cantor and band spetra for periodic quantum graphs with magnetic fields, Comm. Math. Phys., 269 (2007), 87-105. · Zbl 1113.81053 [3] Do, Ngoc T., On the quantum graph spectra of graphyne nanotubes, Anal. Math. Phys., 5 (2015), 39-65. · Zbl 1317.35147 [4] Do, Ngoc T. and Kuchment, P., Quantum graph spectra of a graphyne structure, Nanoscale Syst., 2 (2013), 107-123. · Zbl 1273.81067 [5] Duclos, P., Exner, P. and Turek, O., On the spectrum of a bent chain graph, J. Phys. A: Math. Theor., 41 (2008) 415206(18pp). · Zbl 1192.81167 [6] Exner, P., Lattice Kronig-Penney models, Phys. Rev. Lett., 74 (1995), 3503-3506. [7] Exner, P. and Turek, O., Spectrum of a dilated honeycomb network, Integral Equations and Operator Thoery, 81 (2015), 535-557. · Zbl 1318.81027 [8] Korotyaev, E., Effective masses for zigzag nanotubes in magnetic fields, Lett. Math. Phys., 83 (2008), 83-95. · Zbl 1136.81370 [9] Korotyaev, E. and Lobanov, I., Schrödinger Operators on Zigzag Nanotubes, Ann. Henri Poincaré, 8 (2007), 1151-1176. · Zbl 1375.81098 [10] Kostrykin, V. and Schrader, R., Kirchhoff’s rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630. · Zbl 0928.34066 [11] Kuchment, P. and Post, O., On the Spectra of Carbon Nano-Structures, Commun. Math. Phys., 275 (2007), 805-826. · Zbl 1145.81032 [12] Magnus, W. and Winkler, S., Hill’s Equation, Wiley, 1966. · Zbl 0158.09604 [13] Niikuni, H., Schrödinger operators on a periodically broken zigzag carbon nanotube, Proc. Indian Acad. Sci. (Math. Sci.)., 127 (2017), 471-516. · Zbl 1372.34129 [14] Pankrashkin, K., Spectra of Schrödinger operators on equilateral quantum graphs, Lett. Math. Phys., 77 (2006), 139-154. · Zbl 1113.81056 [15] Pankrashkin, K., Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures, J. Math. Anal. Appl., 396 (2012), 640-655. · Zbl 1260.47010 [16] Pauling, L., The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys., 4 (1936), 693-677. [17] Poschel, J. and Trubowitz, E., Inverse Spectral Theory, Academic Press, Orlando, 1987. · Zbl 0623.34001 [18] Reed, M. and Simon, B., Methods of modern mathematical physics, IV. Analysis of operators, Academic Press, New York, 1978. · Zbl 0401.47001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.