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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
4
Pages
76-83
 Section Mathematics Title Quasi-levels of the Hamiltonian for a carbon nanotube Author(-s) Morozova L.E.a, Chuburin Yu.P.b Affiliations Izhevsk State Technical Universitya, Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesb Abstract In the past two decades, carbon nanotubes have been actively investigated in the physics literature, because of the promising prospects for their use in microelectronics; at the same time, interesting mathematical properties of used Hamiltonians, unfortunately, are often overlooked by mathematicians. In this paper, we carry out the mathematically rigorous investigation of spectral properties of the Hamiltonian $H_{\varepsilon}=H_0+\varepsilon V$, where the Hamiltonian $H_0$ of an electron in a zigzag carbon nanotube is written in the tight-binding approach, and the operator $\varepsilon V$ (potential) has the form $$(\varepsilon V\psi )(n)=\varepsilon { V_1\psi _1(n)\choose V_2\psi _2(n)}\delta_{n0}$$ (here $\varepsilon >0$, $V_1,V_2$ are real numbers, $\delta_{n0}$ is the Kronecker delta). The Hamiltonian $H_{\varepsilon}$ corresponds to the carbon nanotube with an impurity uniformly distributed over the cross section of the nanotube. This Hamiltonian is the difference operator defined on functions from $(l^2(\Omega ))^2$, where $\Omega =\mathbb Z\times \{ 0,1,\ldots,N-1\}$, $N\geqslant 2$, satisfying the periodic boundary conditions. In particular, in this paper we prove that for each subband of the spectrum near one of the boundary points of the subband exactly one quasilevel (i.e. eigenvalue or resonance) exists in the case of small potentials. For quasilevels, the asymptotic formulas of the form $$\lambda _l^{\pm}= \pm \Bigl|2\cos\frac{\pi l}{N}+1\Bigr|\cdot\Bigl(1+\frac{\varepsilon^2(V_1+V_2)^2}{16\cos\frac{\pi l}{N}}\Bigr) +O(\varepsilon^3),$$ are obtained, where $l$ is the subband number, $N$ is the number of atoms in the cross section of the nanotube, and $\pm$ is the sign of the $\lambda$. Also, we find the condition when a quasilevel is an eigenvalue. Keywords Hamiltonian of a carbon nanotube, eigenvalue, resonance UDC 517.958, 530.145.6 MSC 81Q10, 81Q15 DOI 10.20537/vm140406 Received 30 October 2014 Language Russian Citation Morozova L.E., Chuburin Yu.P. Quasi-levels of the Hamiltonian for a carbon nanotube, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 4, pp. 76-83. References Dubois S.M.-M., Zanolli Z., Declerck X., Charliera J.-C. Electronic properties and quantum transport in graphene-based nanostructures, Eur. Phys. J. B., 2009, vol. 72, pp. 1-24. Laird E.A., Kuemmeth F., Steele G., Grove-Rasmussen K., Nygård J., Flensberg K., Kouwenhoven L.P. Quantum transport in carbon nanotubes, 2014, arXiv: 1403.6113 [cond-mat.mes-hall]. http://arxiv.org/pdf/1403.6113v1.pdf Charlier J.-C., Blase X., Roche S. Electronic and transport properties of nanotubes, Rev. Mod. Phys., 2007, vol. 79, pp. 677-732. Reed M. Simon B. Methods of modern mathematical physics. IV. Analysis of operators, New York: Academic Press, 1978. Translated under the title Metody sovremennoi matematicheskoi fiziki. IV. Analiz operatorov, Moscow: Mir, 1982, 428 p. Baranova L.Y., Chuburin Y.P. Quasi-levels of the two-particle discrete Schrödinger operator with a perturbed periodic potential, J. Phys. A.: Math. Theor., 2008, vol. 41, 435205 (11 p). Gunning R., Rossi H. Analytic functions of several complex variables, New York: Prentice-Hall, 1965. Translated under the title Analiticheskie funktsii mnogikh kompleksnykh peremennykh, Moscow: Mir, 1969, 396 p. Full text