no. 5
Eigenvalue asymptotics for Schrödinger operators on sparse graphs
[Asymptotique des valeurs propres pour les opérateurs de Schrödinger agissant sur des graphes éparses]
Bonnefont, Michel1 ; Golénia, Sylvain1 ; Keller, Matthias2
1 Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération F-33405 Talence cedex (France)
2 Friedrich Schiller Universität Jena Mathematisches Institut 07745 Jena (Germany)
Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 1969-1998.

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.

We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics.

DOI : 10.5802/aif.2979
Classification : 47A10, 34L20, 05C63, 47B25, 47A63
Keywords: discrete Laplacian, locally finite graphs, eigenvalues, asymptotic, planarity, sparse, functional inequality
Mot clés : Laplacien discret, graphe locallement fini, valeurs propres, asymptotique, planarité, éparse, inégalité fonctionelle

Bonnefont, Michel 1 ; Golénia, Sylvain 1 ; Keller, Matthias 2

1 Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération F-33405 Talence cedex (France)
2 Friedrich Schiller Universität Jena Mathematisches Institut 07745 Jena (Germany) 
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Bonnefont, Michel; Golénia, Sylvain; Keller, Matthias. Eigenvalue asymptotics for Schrödinger operators on sparse graphs. Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 1969-1998. doi : 10.5802/aif.2979. https://aif.centre-mersenne.org/articles/10.5802/aif.2979/

[1] Alon, Noga; Angel, Omer; Benjamini, Itai; Lubetzky, Eyal Sums and products along sparse graphs, Israel J. Math., Volume 188 (2012), pp. 353-384 | DOI | Zbl

[2] Bauer, Frank; Hua, Bobo; Jost, Jürgen The dual Cheeger constant and spectra of infinite graphs, Adv. Math., Volume 251 (2014), pp. 147-194 | DOI | Zbl

[3] Bauer, Frank; Jost, Jürgen; Liu, Shiping Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator, Math. Res. Lett., Volume 19 (2012) no. 6, pp. 1185-1205 | DOI | Zbl

[4] Bauer, Frank; Keller, Matthias; Wojciechowski, Radosław K. Cheeger inequalities for unbounded graph Laplacians, J. Eur. Math. Soc. (JEMS), Volume 17 (2015) no. 2, pp. 259-271 | DOI

[5] Breuer, Jonathan Singular continuous spectrum for the Laplacian on certain sparse trees, Comm. Math. Phys., Volume 269 (2007) no. 3, pp. 851-857 | DOI | Zbl

[6] Dodziuk, J.; Kendall, W. S. Combinatorial Laplacians and isoperimetric inequality, From local times to global geometry, control and physics (Coventry, 1984/85) (Pitman Res. Notes Math. Ser.), Volume 150, Longman Sci. Tech., Harlow, 1986, pp. 68-74 | Zbl

[7] Dodziuk, Jozef Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., Volume 284 (1984) no. 2, pp. 787-794 | DOI | Zbl

[8] Dodziuk, Józef Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ, 2006, pp. 353-368 | Zbl

[9] Dodziuk, Józef; Mathai, Varghese Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians, The ubiquitous heat kernel (Contemp. Math.), Volume 398, Amer. Math. Soc., Providence, RI, 2006, pp. 69-81 | DOI | Zbl

[10] Erdős, P.; Graham, R. L.; Szemeredi, E. On sparse graphs with dense long paths, Computers and mathematics with applications, Pergamon, Oxford, 1976, pp. 365-369 | Zbl

[11] Fujiwara, Koji The Laplacian on rapidly branching trees, Duke Math. J., Volume 83 (1996) no. 1, pp. 191-202 | DOI | Zbl

[12] Golénia, Sylvain Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians, J. Funct. Anal., Volume 266 (2014) no. 5, pp. 2662-2688 | DOI | Zbl

[13] Higuchi, Yusuke Combinatorial curvature for planar graphs, J. Graph Theory, Volume 38 (2001) no. 4, pp. 220-229 | DOI | Zbl

[14] Jost, J.; Liu, S. Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs (2011) (http://arxiv.org/abs/1103.4037v2)

[15] Keller, M.; Lenz, D. Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom., Volume 5 (2010) no. 4, pp. 198-224 | DOI | Zbl

[16] Keller, M.; Schmidt, M. A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs (2012) (http://arxiv.org/abs/1301.1304)

[17] Keller, Matthias The essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann., Volume 346 (2010) no. 1, pp. 51-66 | DOI | Zbl

[18] Keller, Matthias Curvature, geometry and spectral properties of planar graphs, Discrete Comput. Geom., Volume 46 (2011) no. 3, pp. 500-525 | DOI | Zbl

[19] Keller, Matthias; Lenz, Daniel Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine Angew. Math., Volume 666 (2012), pp. 189-223 | DOI | Zbl

[20] Keller, Matthias; Lenz, Daniel; Wojciechowski, Radosław K. Volume growth, spectrum and stochastic completeness of infinite graphs, Math. Z., Volume 274 (2013) no. 3-4, pp. 905-932 | DOI | Zbl

[21] Keller, Matthias; Peyerimhoff, Norbert Cheeger constants, growth and spectrum of locally tessellating planar graphs, Math. Z., Volume 268 (2011) no. 3-4, pp. 871-886 | DOI | Zbl

[22] Lee, Audrey; Streinu, Ileana Pebble game algorithms and sparse graphs, Discrete Math., Volume 308 (2008) no. 8, pp. 1425-1437 | DOI | Zbl

[23] Lin, Yong; Yau, Shing-Tung Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett., Volume 17 (2010) no. 2, pp. 343-356 | DOI | Zbl

[24] Loréa, M. On matroidal families, Discrete Math., Volume 28 (1979) no. 1, pp. 103-106 | DOI | Zbl

[25] Mohar, Bojan Isoperimetric inequalities, growth, and the spectrum of graphs, Linear Algebra Appl., Volume 103 (1988), pp. 119-131 | DOI | Zbl

[26] Mohar, Bojan Some relations between analytic and geometric properties of infinite graphs, Discrete Math., Volume 95 (1991) no. 1-3, pp. 193-219 Directions in infinite graph theory and combinatorics (Cambridge, 1989) | DOI | Zbl

[27] Mohar, Bojan Many large eigenvalues in sparse graphs, European J. Combin., Volume 34 (2013) no. 7, pp. 1125-1129 | DOI | Zbl

[28] Reed, Michael; Simon, Barry Methods of modern mathematical physics. I, II and IV. Functional analysis, Fourier, Self-adjointness, Academic Press, New York-London, 1975 | Zbl

[29] Stollmann, Peter; Voigt, Jürgen Perturbation of Dirichlet forms by measures, Potential Anal., Volume 5 (1996) no. 2, pp. 109-138 | DOI | Zbl

[30] Weidmann, Joachim Lineare Operatoren in Hilberträumen. Teil 1, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 2000, pp. 475 (Grundlagen. [Foundations]) | DOI | Zbl

[31] Woess, Wolfgang A note on tilings and strong isoperimetric inequality, Math. Proc. Cambridge Philos. Soc., Volume 124 (1998) no. 3, pp. 385-393 | DOI | Zbl

[32] Wojciechowski, Radoslaw Krzysztof Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, 2008, pp. 87 Thesis (Ph.D.)–City University of New York

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