Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T06:16:14.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Schrödinger Operator on the Zigzag Half-Nanotube in MagneticField

Published online by Cambridge University Press:  12 May 2010

A. Iantchenko  and
E. Korotyaev
A. Iantchenko*
Affiliation:
Institute of Mathematics and Physics, Aberystwyth Univ., Penglais, Ceredigion, SY23 3BZ, UK
E. Korotyaev
Affiliation:
School of Mathematics, Cardiff Univ., Senghennydd Road, Cardiff, CF24 4AG, UK
*
* Corresponding author. E-mail:ai@mah.se
Get access

Abstract

We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magneticfield which is described by the magnetic Schrödinger operator with a periodic potentialplus a finitely supported perturbation. We describe all eigenvalues and resonances of thisoperator, and theirs dependence on the magnetic field. The proof is reduced to theanalysis of the periodic Jacobi operators on the half-line with finitely supportedperturbations.

Keywords

nanotubesJacobi operatorperiodicfinite support perturbationresonances
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 175 - 197
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avron, J.E., Raveh, A., Zur, B.. Adiabatic quantum transport in multiply connected systems . Rev. Modern Phys., 60 (1988), No. 4, 873–915.CrossRefGoogle Scholar
Exner, P.. A duality between Schrödinger operators on graphs and certain Jacobi matrices . Ann. Inst. H. Poincaré Phys. Theor., 66 (1997), No. 4, 359–371. Google Scholar
P. Harris. Carbon Nanotubes and Related Structures. Cambridge Univ. Press., Cambridge, 1999.
A. Iantchenko, E. Korotyaev. Periodic Jacobi operators with finitely supported perturbations on the half-line. Preprint, 2009.
Iijima, S.. Helical microtubules of graphitic carbon . Nature, 354 (1991), 56–58.CrossRefGoogle Scholar
Korotyaev, E.. Effective masses for zigzag nanotubes in magnetic fields . Lett. Math. Phys., 83 (2008), No 1, 83–95. CrossRefGoogle Scholar
E. Korotyaev. Resonances for Schrödinger operator with periodic plus compactly supported potentials on the half-line. Preprint, 2008.
E. Korotyaev, A. Kutsenko. Zigzag nanoribbons in external electric Fields. To appear in Asympt. Anal.
E. Korotyaev, A. Kutsenko. Zigzag and armchair nanotubes in external fields. To appear in Diff. Equations: Systems, Applications and Analysis. Nova Science Publishers, Inc.
Korotyaev, E., Lobanov, I.. Schrödinger operators on zigzag periodic graphs . Ann. Henri Poincaré, 8 (2007), 1151–1176.CrossRefGoogle Scholar
E. Korotyaev, I. Lobanov. Zigzag periodic nanotube in magnetic field. Preprint, 2006.
Kuchment, P., Post, O.. On the spectra of carbon nano-structures . Commun. Math. Phys., 275 (2007), 805–826.CrossRefGoogle Scholar
van Moerbeke, P.. The spectrum of Jacobi matrices . Invent. Math., 37 (1976), No. 1, 45–81. CrossRefGoogle Scholar
Novikov, D.S.. Electron properties of carbon nanotubes in a periodic potential . Physical Rev., B 72 (2005), 235428-1-22. CrossRefGoogle Scholar
Pauling, L.. The diamagnetic anisotropy of aromatic molecules . J. of Chem. Phys., 4 (1936), 673–677.CrossRefGoogle Scholar
Pankrashkin, K.. Spectra of Schrödinger operators on equilateral quantum graphs . Lett. Math. Phys., 77 (2006), 139–154.CrossRefGoogle Scholar
Rabinovich, V., Roch, S.. Essential spectra of difference operators on Zn-periodic graphs . J. Phys. A: Math. Theor., 40 (2007), 10109.CrossRefGoogle Scholar
Ruedenberg, K., Scherr, C.W.. Free-electron network model for conjugated systems. I. Theory . J. of Chem. Phys., 21 (1953), 1565–1581.CrossRefGoogle Scholar
R. Saito, G. Dresselhaus, M. Dresselhaus. Physical properties of carbon nanotubes. Imperial College Press, 1998.
G. Teschl. Jacobi operators and completely integrable nonlinear lattices. Providence, RI: AMS, (2000) ( Math. Surveys Monographs, V. 72.)
E.B. Vinberg.A Course in Algebra. Graduate studies in Mathematics, AMS, V. 56.