On the spectra of carbon nano-structures. (English) Zbl 1145.81032

Summary: An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided.


81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q99 General mathematical topics and methods in quantum theory
81U30 Dispersion theory, dispersion relations arising in quantum theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
82D20 Statistical mechanics of solids
82D99 Applications of statistical mechanics to specific types of physical systems
Full Text: DOI arXiv


[1] Alexander S. (1985). Superconductivity of networks. A percolation approach to the effects of disorder. Phys. Rev. B 27: 1541–1557
[2] Amovilli C., Leys F. and March N. (2004). Electronic energy spectrum of two-dimensional solids and a chain of C atoms from a quantum network model. J. Math. Chem. 36(2): 93–112 · Zbl 1052.81689
[3] Amovilli C., Leys F. and March N. (2004). Topology, connectivity, and electronic structure of C and B cages and the corresponding nanotubes. J. Chem. Inf. Comput. Sci. 44: 122–135
[4] Ashoft N.W. and Mermin N.D. (1976). Solid State Physics. New York-London, Holt, Rinehart and Winston
[5] Avron J., Raveh A. and Zur B. (1988). Adiabatic quantum transport in multiply connected systems. Rev. Mod. Phys. 60(4): 873–915
[6] Badanin, A., Brüning, J., Korotyaev, E., Lobanov, I.: Schrödinger operators on armchair nanotubes. Preprint, (Dec 27th 2006)
[7] Berkolaiko, G., Carlson, R., Fulling, S., Kuchment, P.: (eds): Quantum Graphs and Their Applications, Contemp. Math. 415, Providence, RI: Amer. Math. Soc. 2006 · Zbl 1098.81007
[8] Cattaneo C. (1997). The spectrum of the continuous Laplacian on a graph. Monatsh. Math. 124(3): 215–235 · Zbl 0892.47001
[9] Chung, F.: Spectral Graph Theory. Providence RI: Amer. Math. Soc., 1997
[10] Colin de Verdière, Y.: Spectres De Graphes. Paris: Societe Mathematique De France, 1998 · Zbl 0913.05071
[11] de Gennes P.-G. (1981). Champ itique d’une boucle supraconductrice ramefiee. C. R. Acad. Sc. Paris 292B: 279–282
[12] Djakov P. and Mityagin B.S. (2006). Instability zones of periodic 1-dimensional Schrödinger and Dirac operators. Russ. Math. Surv. 61(4): 663–766 · Zbl 1128.47041
[13] Duclos P. and Exner P. (1995). Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7: 73–102 · Zbl 0837.35037
[14] Exner P. (1996). Contact interactions on graph superlattices. J. Phys. A29: 87–102 · Zbl 0916.47056
[15] Exner P. and Gawlista R. (1996). Band spectra of rectangular graph superlattices. Phys. Rev. B53: 7275–7286
[16] Exner, P., Seba, P.: Electrons in semiconductor miostructures: a challenge to operator theorists. In: Proceedings of the Workshop on Schrödinger Operators, Standard and Nonstandard (Dubna 1988), Singapore: World Scientific, 1989 pp. 79–100
[17] Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. Edinburgh-London: Scottish Acad. Press Ltd., 1973 · Zbl 0287.34016
[18] Garnett J. and Trubowitz E. (1984). Gaps and bands of one-dimensional periodic Schrödinger operators. Comment. Math. Helv. 59(2): 258–312 · Zbl 0554.34013
[19] Garnett J. and Trubowitz E. (1987). Gaps and bands of one dimensional periodic Schrödinger operators II. Comment. Math. Helv. 62: 18–37 · Zbl 0649.34034
[20] Gerard C. and Nier F. (1998). The Mourre theory for analytically fibered operators. J. Funct. Anal. 152(1): 202–219 · Zbl 0939.47019
[21] Harris P. (2002). Carbon Nano-tubes and Related Structures. Cambridge University Press, Cambridge
[22] Hochstadt H. (1963). Estimates on the stability intervals for the Hill’s equation. Proc. AMS 14: 930–932 · Zbl 0122.09202
[23] Hochstadt H. (1968). On the determination of a Hill’s equation from its spectrum. Arch. Rat. Mech. Anal. 19: 353–362 · Zbl 0128.31201
[24] Iakubovich V.A. and Starzhinski V.M. (1975). Linear Differential Equations with Periodic Coefficients. Wiley, NY
[25] Katsnelson M.I. (2007). Graphene: carbon in two dimensions. Materials Today 10(1–2): 20–27
[26] Korotyaev, E., Lobanov, I.: Schrödinger operators on zigzag graphs. http://arxiv.org/list/math.SP/ 0604006, 2006 · Zbl 1375.81098
[27] Korotyaev, E., Lobanov, I.: Zigzag periodic nanotube in magnetic field. http://arxiv.org/list/math.SP/ 0604007, 2006
[28] Kostrykin V. and Schrader R. (1999). Kirchhoff’s rule for quantum wires. J. Phys. A 32: 595–630 · Zbl 0928.34066
[29] Kottos T. and Smilansky U. (1997). Quantum chaos on graphs. Phys. Rev. Lett. 79: 4794–4797
[30] Kuchment, P.: To the Floquet theory of periodic difference equations. In: Geometrical and Algebraical Aspects in Several Complex Variables, Cetraro (Italy), June 1989, Carouge: EditEl, 1991, pp 203–209
[31] Kuchment P. (1993). Floquet Theory for Partial Differential Equations. Birkhäuser Verlag, Basel · Zbl 0789.35002
[32] (2004). Quantum graphs and their applications. Special issue, Waves in Random Media
[33] Kuchment P. (2002). Graph models of wave propagation in thin structures. Waves in Random Media 12(4): R1–R24 · Zbl 1063.35525
[34] Kuchment P. (2004). Quantum graphs: I. Some basic structures. Waves Random Media 14: S107–S128 · Zbl 1063.81058
[35] Kuchment P. (2005). Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A 38(22): 4887–4900 · Zbl 1070.81062
[36] Kuchment P. and Kunyansky L. (1999). Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs. Exp. Math. 8(1): 1–28 · Zbl 0930.35112
[37] Kuchment P. and Pinchover Y. (2001). Integral representations and Liouville theorems for solutions of periodic elliptic equations. J. Funct. Anal. 181: 402–446 · Zbl 0986.35028
[38] Kuchment, P., Pinchover, Y.: Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds. http://arxiv.org/list/math-ph/0503010, 2005 to appear in Trans. Amer. Math. Soc. · Zbl 1127.58015
[39] Kuchment P. and Vainberg B. (2006). On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators. Commun. Math. Phys. 268: 673–686 · Zbl 1125.39021
[40] Lazutkin V.F. and Pankratova T.F. (1974). Asymptotics of the width of gaps in the spectrum of the Sturm-Liouville operators with periodic potential. Soviet Math. Dokl. 15: 649–653 · Zbl 0318.34035
[41] Magnus W. and Winkler S. (1966). Hill’s Equation. Wiley, NY · Zbl 0158.09604
[42] Marchenko, V.A., Ostrovskii, I.V.: A characterization of the spectrum of Hill’s operator. Matem. Sborn. 97, 540–606 (1975); English transl. in Math. USSR-Sb. 26, 493–554 (1975) · Zbl 0327.34021
[43] Marchenko, V.A., Ostrovskii, I.V.: Approximation of periodic potentials by finite zone potentials. (Russian) Vestnik Kharkov. Gos. Univ. No. 205, 4–40, 139 (1980)
[44] McKean H.P. and Trubowitz E. (1978). Hill’s surfaces and their theta functions. Bull. Amer. Math. Soc. 84(6): 1042–1085 · Zbl 0428.34026
[45] Mills R.G.J. and Montroll E.W. (1970). Quantum theory on a network. II. A solvable model which may have several bound states per node point. J. Math. Phys. 11(8): 2525–2538
[46] Molchanov, S., Vainberg, B.: Transition from a network of thin fibers to the quantum graph: an explicitly solvable model. Cont. Math. 415, Providence, RI: Amer. Math. Soc., 2006, pp 227–240 · Zbl 1317.35046
[47] Molchanov, S., Vainberg, B.: Scattering solutions in a network of thin fibers: small diameter asymptotics. http://arixiv.org/list/math-ph/0609021, 2006 · Zbl 1210.35036
[48] Montroll E. (1970). Quantum theory on a network. I. A solvable model whose wavefunctions are elementary functions. J. Math. Phys. 11(2): 635–648
[49] Oleinik V.L., Pavlov B.S. and Sibirev N.V. (2004). Analysis of the dispersion equation for the Schrödinger operator on periodic metric graphs. Waves in Random Media 14: 157–183 · Zbl 1076.47032
[50] Pankrashkin K. (2006). Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77: 139–154 · Zbl 1113.81056
[51] Pauling L. (1936). The diamagnetic anisotropy of aromatic molecules. J. Chem. Phys. 4: 673–677
[52] Post O. (2005). Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case. J. Phys. A 38(22): 4917–4931 · Zbl 1072.81019
[53] Reed M. and Simon B. (1978). Methods of modern mathematical physics IV: Analysis of operators. Academic Press, New York · Zbl 0401.47001
[54] Ruedenberg K. and Scherr C.W. (1953). Free-electron network model for conjugated systems. I. Theory. J. Chem. Phys., 21(9): 1565–1581
[55] Saito R., Dresselhaus G. and Dresselhaus M.S. (1998). Physical Properties of Carbon Nanotubes. Imperial College Press, London · Zbl 0752.68069
[56] Thomas L.E. (1973). Time dependent approach to scattering from impurities in a crystal. Comm. Math. Phys. 33: 335–343
[57] Trubowitz E. (1977). The inverse problem for periodic potentials, Comm. Pure and Appl. Math. 30: 321–342 · Zbl 0403.34022
[58] Zakharov V.E., Manakov S.V., Novikov S.P. and Pitaevskii L.P. (1984). Theory of Solitons: The Inverse Scattering Method. Plenum, London · Zbl 0598.35002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.