Comptes Rendus
Asymptotics of the spectrum of the Dirichlet Laplacian on a thin carbon nano-structure
Comptes Rendus. Mécanique, Volume 343 (2015) no. 5-6, pp. 360-364.

For the honeycomb lattice of quantum waveguides, the limit passage is performed when the relative thickness h of ligaments tends to zero and the asymptotic structure of the spectrum of the Dirichlet Laplacian is described.

Pour la structure en nid d'abeille du guide d'ondes quantique, on réalise un passage à la limite lorsque l'épaisseur relative h des liaisons tend vers zéro, et on décrit le comportement asymptotique du spectre de l'opérateur laplacien de Dirichlet.

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Published online:
DOI: 10.1016/j.crme.2015.03.001
Keywords: Spectrum, Dirichlet problem, Asymptotics, Quantum waveguide, Trapped modes and bounded solutions
Mot clés : Spectre, Opérateur laplacien de Dirichlet, Analyse asymptotique, Guides d'ondes quantiques, Modes piégés et solutions bornées

Sergei A. Nazarov 1, 2; Keijo Ruotsalainen 3; Pauliina Uusitalo 3

1 Saint-Petersburg State University, Universitetsky pr., 28, Peterhof, St. Petersburg, 198504, Russia
2 Saint-Petersburg State Polytechnical University, Polytechnicheskaya ul., 29, St. Petersburg, 195251, Russia
3 University of Oulu, Mathematics Division, P.O. Box 4500, 90014, Oulu, Finland
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Sergei A. Nazarov; Keijo Ruotsalainen; Pauliina Uusitalo. Asymptotics of the spectrum of the Dirichlet Laplacian on a thin carbon nano-structure. Comptes Rendus. Mécanique, Volume 343 (2015) no. 5-6, pp. 360-364. doi : 10.1016/j.crme.2015.03.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.03.001/

[1] M.M. Skriganov Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proc. Steklov Inst. Math., Volume 171 (1987) no. 2 (vi–121)

[2] P.A. Kuchment Floquet Theory for Partial Differential Equations, Birkhäuser, Basel, Switzerland, 1993

[3] K.S. Novoselov; A.K. Geim; S.V. Morozov; D. Jiang; Y. Zhang; S.V. Dubonos; I.V. Grigorieva; A.A. Firsov Electric field in atomically thin carbon films, Science, Volume 306 (2004) no. 5696, pp. 666-669

[4] P.A. Kuchment Graph models for waves in thin structures, Waves Random Media, Volume 12 (2002) no. 4, p. R1-R24

[5] P.A. Kuchment; O. Post On the spectra of carbon nano-structure, Commun. Math. Phys., Volume 275 (2007) no. 3, pp. 805-826

[6] L. Pauling The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys., Volume 4 (1936), pp. 673-677

[7] D. Grieser Spectra of graph neighborhoods and scattering, Proc. Lond. Math. Soc., Volume 97 (2008) no. 3, pp. 718-752

[8] P. Exner; O. Post Convergence of spectra of graph-like thin manifolds, J. Geom. Phys., Volume 54 (2005) no. 1, pp. 77-115

[9] Y. Saito The limiting equation for Neumann Laplacians on shrinking domains, Electron. J. Differ. Equ., Volume 31 (2000), pp. 1-25

[10] J. Rubinstein; M. Schatzman Variational problems on multiply connected thin strips I: Basic estimates and convergence of the Laplacian spectrum, Arch. Ration. Mech. Anal., Volume 160 (2001) no. 4, pp. 271-308

[11] P.A. Kuchment; H. Zeng Convergence of spectra of mesoscopic systems collapsing onto a graph, J. Math. Anal. Appl., Volume 258 (2001), pp. 671-700

[12] O. Post Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case, J. Phys. A, Math. Gen., Volume 38 (2005) no. 22, pp. 4917-4931

[13] J. Wyndham The Day of the Triffids, Michael Joseph, 1951

[14] S.A. Nazarov; K. Ruotsalainen; P. Uusitalo The Y-junction of quantum waveguides, Z. Angew. Math. Mech., Volume 94 (2014) no. 6, pp. 477-486

[15] R. Leis Initial Boundary Value Problems of Mathematical Physics, B.G. Teubner, Stuttgart, Germany, 1986

[16] M.G. Lamé Leçons sur la Théorie Mathématique de l'Élasticité des Corps Solides, Gauthier-Villars, Paris, 1866

[17] M.S. Birman; M.Z. Solomyak Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, the Netherlands, 1987

[18] S.A. Nazarov Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide, Theor. Math. Phys., Volume 167 (2011) no. 2, pp. 606-627

[19] S.A. Nazarov Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold, Sib. Math. J., Volume 51 (2010) no. 5, pp. 866-878

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