logo ANNALES DE L'INSTITUT FOURIER

Avec cedram.org

Table des matières de ce fascicule | Article précédent | Article suivant
Michel Bonnefont; Sylvain Golénia; Matthias Keller
Eigenvalue asymptotics for Schrödinger operators on sparse graphs
(Asymptotique des valeurs propres pour les opérateurs de Schrödinger agissant sur des graphes éparses)
Annales de l'institut Fourier, 65 no. 5 (2015), p. 1969-1998, doi: 10.5802/aif.2979
Article PDF
Class. Math.: 47A10, 34L20, 05C63, 47B25, 47A63
Mots clés: Laplacien discret, graphe locallement fini, valeurs propres, asymptotique, planarité, éparse, inégalité fonctionelle

Résumé - Abstract

Nous considérons des opérateurs de Schrödinger agissant sur des graphes éparses. Le fait d’être éparse est équivalent à une inégalité fonctionnelle pour le Laplacien. En particulier il y a des conséquences spectrales fortes pour le Laplacien quand le graphe est éparse : caractérisation de son domaine de forme et de l’absence du spectre essentiel. Dans ce dernier cas, nous calculons l’asymptotique des valeurs propres.

Bibliographie

[1] Noga Alon, Omer Angel, Itai Benjamini & Eyal Lubetzky, “Sums and products along sparse graphs”, Israel J. Math. 188 (2012), p. 353-384 Article |  Zbl 1288.05124
[2] Frank Bauer, Bobo Hua & Jürgen Jost, “The dual Cheeger constant and spectra of infinite graphs”, Adv. Math. 251 (2014), p. 147-194 Article |  Zbl 1285.05133
[3] Frank Bauer, Jürgen Jost & Shiping Liu, “Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator”, Math. Res. Lett. 19 (2012) no. 6, p. 1185-1205 Article |  Zbl 1297.05143
[4] Frank Bauer, Matthias Keller & Radosław K. Wojciechowski, “Cheeger inequalities for unbounded graph Laplacians”, J. Eur. Math. Soc. (JEMS) 17 (2015) no. 2, p. 259-271 Article
[5] Jonathan Breuer, “Singular continuous spectrum for the Laplacian on certain sparse trees”, Comm. Math. Phys. 269 (2007) no. 3, p. 851-857 Article |  Zbl 1120.39023
[6] J. Dodziuk & W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality, From local times to global geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser. 150, Longman Sci. Tech., Harlow, 1986, p. 68–74  Zbl 0619.05005
[7] Jozef Dodziuk, “Difference equations, isoperimetric inequality and transience of certain random walks”, Trans. Amer. Math. Soc. 284 (1984) no. 2, p. 787-794 Article |  Zbl 0512.39001
[8] Józef Dodziuk, Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ, 2006, p. 353–368  Zbl 1127.58034
[9] Józef Dodziuk & Varghese Mathai, Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians, The ubiquitous heat kernel, Contemp. Math. 398, Amer. Math. Soc., Providence, RI, 2006, p. 69–81 Article |  Zbl 1207.81024
[10] P. Erdős, R. L. Graham & E. Szemeredi, On sparse graphs with dense long paths, Computers and mathematics with applications, Pergamon, Oxford, 1976, p. 365–369  Zbl 0328.05123
[11] Koji Fujiwara, “The Laplacian on rapidly branching trees”, Duke Math. J. 83 (1996) no. 1, p. 191-202 Article |  Zbl 0856.58044
[12] Sylvain Golénia, “Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians”, J. Funct. Anal. 266 (2014) no. 5, p. 2662-2688 Article |  Zbl 1292.35300
[13] Yusuke Higuchi, “Combinatorial curvature for planar graphs”, J. Graph Theory 38 (2001) no. 4, p. 220-229 Article |  Zbl 0996.05041
[14] J. Jost & S. Liu, “Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs”, http://arxiv.org/abs/1103.4037v2, 2011
[15] M. Keller & D. Lenz, “Unbounded Laplacians on graphs: basic spectral properties and the heat equation”, Math. Model. Nat. Phenom. 5 (2010) no. 4, p. 198-224 Article |  Zbl 1207.47032
[16] M. Keller & M. Schmidt, “A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs”, http://arxiv.org/abs/1301.1304, 2012
[17] Matthias Keller, “The essential spectrum of the Laplacian on rapidly branching tessellations”, Math. Ann. 346 (2010) no. 1, p. 51-66 Article |  Zbl 1285.05115
[18] Matthias Keller, “Curvature, geometry and spectral properties of planar graphs”, Discrete Comput. Geom. 46 (2011) no. 3, p. 500-525 Article |  Zbl 1228.05129
[19] Matthias Keller & Daniel Lenz, “Dirichlet forms and stochastic completeness of graphs and subgraphs”, J. Reine Angew. Math. 666 (2012), p. 189-223 Article |  Zbl 1252.47090
[20] Matthias Keller, Daniel Lenz & Radosław K. Wojciechowski, “Volume growth, spectrum and stochastic completeness of infinite graphs”, Math. Z. 274 (2013) no. 3-4, p. 905-932 Article |  Zbl 1269.05051
[21] Matthias Keller & Norbert Peyerimhoff, “Cheeger constants, growth and spectrum of locally tessellating planar graphs”, Math. Z. 268 (2011) no. 3-4, p. 871-886 Article |  Zbl 1250.05039
[22] Audrey Lee & Ileana Streinu, “Pebble game algorithms and sparse graphs”, Discrete Math. 308 (2008) no. 8, p. 1425-1437 Article |  Zbl 1136.05062
[23] Yong Lin & Shing-Tung Yau, “Ricci curvature and eigenvalue estimate on locally finite graphs”, Math. Res. Lett. 17 (2010) no. 2, p. 343-356 Article |  Zbl 1232.31003
[24] M. Loréa, “On matroidal families”, Discrete Math. 28 (1979) no. 1, p. 103-106 Article |  Zbl 0409.05050
[25] Bojan Mohar, “Isoperimetric inequalities, growth, and the spectrum of graphs”, Linear Algebra Appl. 103 (1988), p. 119-131 Article |  Zbl 0658.05055
[26] Bojan Mohar, “Some relations between analytic and geometric properties of infinite graphs”, Discrete Math. 95 (1991) no. 1-3, p. 193-219, Directions in infinite graph theory and combinatorics (Cambridge, 1989) Article |  Zbl 0801.05051
[27] Bojan Mohar, “Many large eigenvalues in sparse graphs”, European J. Combin. 34 (2013) no. 7, p. 1125-1129 Article |  Zbl 1292.05178
[28] Michael Reed & Barry Simon, Methods of modern mathematical physics. I, II and IV. Functional analysis, Fourier, Self-adjointness, Academic Press, New York-London, 1975  Zbl 0242.46001
[29] Peter Stollmann & Jürgen Voigt, “Perturbation of Dirichlet forms by measures”, Potential Anal. 5 (1996) no. 2, p. 109-138 Article |  Zbl 0861.31004
[30] Joachim Weidmann, Lineare Operatoren in Hilberträumen. Teil 1, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 2000, Grundlagen. [Foundations] Article |  Zbl 0344.47001
[31] Wolfgang Woess, “A note on tilings and strong isoperimetric inequality”, Math. Proc. Cambridge Philos. Soc. 124 (1998) no. 3, p. 385-393 Article |  Zbl 0914.05015
[32] Radoslaw Krzysztof Wojciechowski, Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, 2008, Thesis (Ph.D.)–City University of New York
haut