A magnetic Schrödinger operator on a periodic graph

A. V. Badanin   ; E. L. Korotyaev

Sbornik. Mathematics, Tome 201 (2010) no. 10, pp. 1403-1448

Résumé

This paper looks at a magnetic Shrödinger operator on a graph of special form in $\mathbb R^3$. It is called an armchair graph because graphs of this form with operators on them are used as a possible model for the so-called armchair nanotube in the homogeneous magnetic field which has amplitude $b$ and is parallel to the axis of the nanotube. The spectrum of the operator in question consists of an absolutely continuous part (spectral bands, separated by gaps) and finitely many eigenvalues of infinite multiplicity. The asymptotic behaviour of gaps for fixed $b$ and high energies is described; it is proved that for all values of $b$, apart from a discrete set containing $b=0$, there exists an infinite system of nondegenerate gaps $G_n$ with length $|G_n|\to\infty$ as $n\to\infty$. The dependence of the spectrum on the magnetic field is investigated and the existence of gaps independent of $b$ is proved for certain special potentials. The asymptotic behaviour of gaps as $b\to0$ is described. Bibliography: 32 titles.

Détail

Keywords: periodic graph, magnetic Schrödinger operator, spectral bands, asymptotic behaviour of spectral bands.

Bibliographie

[1] R. Saito, G. Dresselhaus, M. Dresselhaus, Physical properties of carbon nanotubes, Imperial College Press, Tokyo, 1998

[2] P. Kharris, Uglerodnye nanotruby i rodstvennye struktury. Novye materialy XXI veka, Tekhnosfera, M., 2003

[3] E. Korotyaev, I. Lobanov, “Schrödinger operators on zigzag nanotubes”, Ann. Henri Poincaré, 8:6 (2007), 1151–1176 | DOI | MR | Zbl

[4] V. A. Marchenko, Sturm–Liouville operators and applications, Oper. Theory Adv. Appl., 22, Birkhäuser, Basel, 1986 | MR | MR | Zbl | Zbl

[5] W. Magnus, S. Winkler, Hill's equation, Dover, New York, 1979 | MR | Zbl

[6] E. Korotyaev, I. Lobanov, Zigzag periodic nanotube in magnetic field, arXiv: math/0604007

[7] A. Badanin, J. Brüning, E. Korotyaev, “The Lyapunov function for Schrodinger operators with a periodic $2\times2$ matrix potential”, J. Funct. Anal., 234:1 (2006), 106–126 | DOI | MR | Zbl

[8] D. Chelkak, E. Korotyaev, “Spectral estimates for Schrodinger operators with periodic matrix potentials on the real line”, Int. Math. Res. Not., 7 (2006), ID 60314 | MR | Zbl

[9] E. Korotyaev, “Conformal spectral theory for the monodromy matrix”, Trans. Amer. Math. Soc., 362:7 (2010), 3435–3462 | DOI | MR | Zbl

[10] A. Badanin, J. Brüning, E. Korotyaev, I. Lobanov, “Schrödinger operator on armchair nanotube. I”, arXiv: 0707.3909

[11] A. Badanin, J. Brüning, E. Korotyaev, “Schrödinger operator on armchair nanotube. II”, arXiv: 0707.3900

[12] L. Pauling, “The diamagnetic anisotropy of aromatic molecules”, J. Chem. Phys., 4 (1936), 673–677 | DOI

[13] K. Ruedenberg, Ch. W. Scherr, “Free-electron network model for conjugated systems. I. Theory”, J. Chem. Phys., 21 (1953), 1565–1581 | DOI

[14] C. A. Coulson, “Note on the applicability of the free-electron network model to metals”, Proc. Phys. Soc. Ser. A, 67:7 (1954), 608–614 | DOI | Zbl

[15] E. W. Montroll, “Quantum theory on a network. I. A solvable model whose wavefunctions are elementary functions”, J. Math. Phys., 11:2 (1970), 635–648 | DOI

[16] S. Novikov, “Schrödinger operators on graphs and symplectic geometry”, The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., 24, Amer. Math. Soc., Providence, RI, 1999, 397–413 | MR | Zbl

[17] V. Kostrykin, R. Schrader, “Quantum wires with magnetic fluxes”, Comm. Math. Phys., 237:1–2 (2003), 161–179 | DOI | MR | Zbl

[18] K. Pankrashkin, “Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction”, J. Math. Phys., 47:11 (2006), 112105 | DOI | MR | Zbl

[19] J. Brüning, V. Geyler, K. Pankrashkin, “Cantor and band spectra for periodic quantum graphs with magnetic fields”, Comm. Math. Phys., 269:1 (2007), 87–105 | DOI | MR | Zbl

[20] V. S. Rabinovich, S. Roch, “Essential spectra of difference operators on ${\mathbb Z}^n$-periodic graphs”, J. Phys. A, 40:33 (2007), 10109–10128 | DOI | MR | Zbl

[21] A. Jorio, G. Dresselhaus, M. S. Dresselhaus, Carbon nanotubes: advanced topics in the synthesis, structure, properties and applications, Springer-Verlag, Berlin, Heidelberg, 2007

[22] J.-C. Charlier, P. C. Eklund, J. Zhu, A. C. Ferrari, “Electron and phonon properties of graphene: their relationship with carbon nanotubes”, Carbon nanotubes, Topics in Applied Physics, 111, Springer-Verlag, Berlin, 2008, 673–709 | DOI

[23] T. Ando, “The electronic properties of graphene and carbon nanotubes”, NPG Asia Mater, 1:1 (2009), 17–21

[24] E. L. Korotyaev, A. Kutsenko, “Zigzag nanoribbons in external electric fields”, Asymptot. Anal., 66:3–4 (2010), 187–206 | DOI | MR | Zbl

[25] E. Korotyaev, A. Kutsenko, “Zigzag and armchair nanotubes in external fields”, Differential equations: systems, applications and analysis, Nova Science Publ., 2009, arXiv: 0906.4018

[26] E. Korotyaev, “Effective masses for zigzag nanotubes in magnetic fields”, Lett. Math. Phys., 83:1 (2008), 83–95 | DOI | MR | Zbl

[27] K. Pankrashkin, “Spectra of Schrödinger operators on equilateral quantum graphs”, Lett. Math. Phys., 77:2 (2006), 139–154 | DOI | MR | Zbl

[28] P. Kuchment, O. Post, “On the spectra of carbon nano-structures”, Comm. Math. Phys., 275:3 (2007), 805–826 | DOI | MR | Zbl

[29] M. G. Krejn, “Foundations of the theory of lambda-zones of stability of a canonical system of linear differential equations with periodic coefficients”, Amer. Math. Soc. Transl. 2, 120 (1983), 1–70 | MR | Zbl | Zbl

[30] Ch. Gerard, F. Nier, “The Mourre theory for analytically fibered operators”, J. Funct. Anal., 152:1 (1998), 202–219 | DOI | MR | Zbl

[31] W. A. Harrison, Electronic structure and the properties of solids: The physics of the chemical band, Dover, New York, 1989

[32] P. Kargaev, E. Korotyaev, “The inverse problem for the Hill operator, a direct approach”, Invent. Math., 129:3 (1997), 567–593 | DOI | MR | Zbl