On the Spectra of Carbon Nano-Structures

Kuchment, Peter; Post, Olaf

doi:10.1007/s00220-007-0316-1

Cited by 160 publications

(229 citation statements)

References 48 publications

(149 reference statements)

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“…In order to analyze the spectrum of the Hamiltonians, they gave the unitarily equivalence between the operator and the direct sum of its corresponding Hamiltonians on a quasi-1D periodic metric graph, which has a necklace structure. Namely, they reduced the problem to Hamiltonians on the metric graph consisted of lines and rings (see also [9] for the analysis of the spectrum of carbon nano-structures). The model is called the degenerate zigzag nanotube.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph

Niikuni¹

2015

OpMath

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph

Niikuni¹

2015

OpMath

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“…The observation of the mandatory appearance of Dirac cones in honeycombsymmetric structures was first made (way before the graphene came into play) for the tight-binding model of the discrete honeycomb lattice [392], which provides an approximate picture of the graphene's first two dispersion bands. An infiniteband quantum graph model, which has an infinite-dimensional freedom of choosing honeycomb-symmetric potentials was considered in [255], where the detailed structure of the spectrum and dispersion relation, including in particular the presence of Dirac cones, was described.…”

Section: On a Z 2 -Periodic Graph γ Having Just Two Vertices (Atoms)mentioning

confidence: 99%

“…The Schrödinger operator with honeycomb lattice of point scatterers was considered in [263]. Carbon nanotubes as folded sheets of graphene have been studied in various papers, see, e.g., [106,234,255] and references therein.…”

Section: On a Z 2 -Periodic Graph γ Having Just Two Vertices (Atoms)mentioning

confidence: 99%

Untitled

2016

Bull. Amer. Math. Soc.

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Abstract. The article surveys the main topics, techniques, and results of the theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which significantly influence most spectral features of such operators.The approaches described are applicable not only to the standard model example of Schrödinger operator with periodic electric potential −Δ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases.Important applications are mentioned. However, due to the size restrictions, they are not dealt with in detail.

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“…Major theoretical concern so far, however, is limited to solving stationary states of the linear Schrödinger equation, and to obtaining the energy spectra in closed networks and transmission probabilities for open networks with semi-infinite leads [6][7][8][9][10][11]. Only a few studies treats the nonlinear Schrödinger equation on simple networks, which are still limited to the analysis of its stationary state [12,13].…”

Section: Introductionmentioning

confidence: 99%

Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation

et al. 2011

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We elucidate the case in which the Ablowitz-Ladik (AL) type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple 1-dimensional (1-d) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one (Sobirov et al, Phys. Rev. E 81, 066602 (2010)) on the continuum NLSE to the discrete case. We find: (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1-d discrete chain, but is multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; (3) under findings (1) and (2), there exist an infinite number of constants of motion. As a practical issue, with use of AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.

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On the Spectra of Carbon Nano-Structures

Abstract: Abstract. 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.

Cited by 160 publications

References 48 publications

Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph

Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph

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Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation

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