logo AFST

Essential spectrum and Weyl asymptotics for discrete Laplacians
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 3, pp. 563-624.

Dans cet article, nous étudions le spectre de Laplaciens discrets. Notre travail est basé sur l’inégalité de Hardy et l’étude des fonctions super-harmoniques. Nous retrouvons et améliorons des bornes inférieures pour le bas du spectre et le bas du spectre essentiel. Dans certains cas, nous obtenons des asymptotiques de Weyl pour les valeurs propres. Nous donnons aussi une représentation probabiliste des fonctions super-harmoniques, puis avec des arguments de type couplage, nous établissons des résultats de comparaison pour le bas du spectre, le bas du spectre essentiel et la complétude stochastique de différents Laplaciens discrets. Une classe de graphes faiblement symétriques est aussi étudiée en grand détail.

In this paper, we investigate spectral properties of discrete Laplacians. Our study is based on the Hardy inequality and the use of super-harmonic functions. We recover and improve lower bounds for the bottom of the spectrum and of the essential spectrum. In some situation, we obtain Weyl asymptotics for the eigenvalues. We also provide a probabilistic representation of super-harmonic functions. Using coupling arguments, we set comparison results for the bottom of the spectrum, the bottom of the essential spectrum and the stochastic completeness of different discrete Laplacians. The class of weakly spherically symmetric graphs is also studied in full detail.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1456
Classification : 34L20,  05C63,  47B25,  47A63,  47A10
Mots clés : discrete Laplacian, locally finite graphs, asympotic of eigenvalues, spectrum, essential spectrum, markov chains, functional inequalities
@article{AFST_2015_6_24_3_563_0,
     author = {Michel Bonnefont and Sylvain Gol\'enia},
     title = {Essential spectrum and {Weyl} asymptotics for discrete {Laplacians}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {563--624},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {3},
     year = {2015},
     doi = {10.5802/afst.1456},
     mrnumber = {3403733},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1456/}
}
Michel Bonnefont; Sylvain Golénia. Essential spectrum and Weyl asymptotics for discrete Laplacians. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 3, pp. 563-624. doi : 10.5802/afst.1456. https://afst.centre-mersenne.org/articles/10.5802/afst.1456/

[1] Allegretto (W.) and Huang (Y.X.).— A Picone’s identity for the p-Laplacian an applications. Nonlinear Anal. 32 no. 7, p. 422-438 (1998). | MR 1618334 | Zbl 0930.35053

[2] Bonnefont (M.), Golénia (S.), and Keller (M.).— Eigenvalue asymptotics for Schrödinger operators on sparse graphs, to appear in Annales de l’institut Fourier.

[3] Bauer (F.), Hua (B.), and Jost (J.).— The dual Cheeger constant and spectra of infinite graphs, Adv. in Math., 251, 30, p. 147-194 (2014). | MR 3130339 | Zbl 1285.05133

[4] Breuer (J.) and Keller (M.).— Spectral analysis of certain spherically homogenous graphs, Oper. Matrices 7, no. 4, p. 825-847 (2013). | MR 3154573

[5] Bauer (F.), Keller (M.), and Wojciechowski (R.K.).— Cheeger inequalities for unbounded graph Laplacians, to appear in J. Eur. Math. Soc. (JEMS), arXiv:1209.4911 (2012). | EuDML 277648 | MR 3317744

[6] Cattiaux (P.), Guillin (A.), Wang (F.Y.), and Wu (L.).— Lyapunov conditions for super-Poincaré inequalities, J. Funct. Anal. 256, no. 6, 1 p. 821-1841 (2009). | MR 2498560 | Zbl 1167.26007

[7] Cattiaux (P.), Guillin (A.), and Zitt (P.A.).— Poincaré inequalities and hitting times, Ann. Inst. Henri Poincaré Probab. Stat. 49, no. 1, p. 95-118 (2013). | EuDML 272061 | Numdam | MR 3060149 | Zbl 1270.26018

[8] Colin De Verdière (Y.), Torki-Hamza (N.), and Truc (F.).— Essential self-adjointness for combinatorial Schrödinger operators II- Metrically non complete graphs, Mathematical Physics Analysis and Geometry 14, 1 p. 21-38 (2011). | MR 2782792 | Zbl 1244.05155

[9] Colin De Verdière (Y.), Torki-Hamza (N.), and Truc (F.).— Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields, Ann. Fac. Sci. Toulouse Math. (6) 20, no. 3, p. 599-611 (2011). | EuDML 219819 | Numdam | MR 2894840 | Zbl 1250.47025

[10] Dodziuk (J.).— Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284, no. 2, p. 787-794 (1984). | MR 743744 | Zbl 0512.39001

[11] Dodziuk (J.).— Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ, p. 353-368 (2006). | MR 2246774 | Zbl 1127.58034

[12] Dodziuk (J.) and Kendall (W.S.).— Combinatorial Laplacians and isoperimetric inequality, from local times to global geometry, control and physics (Coventry, 1984/85), p. 68-74, Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow (1986). | MR 894523 | Zbl 0619.05005

[13] Dodziuk (J.) and Matthai (V.).— Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians, The ubiquitous heat kernel, p. 69-81, Contemp. Math., 398, Amer. Math. Soc., Providence, RI (2006.) | MR 2218014 | Zbl 1207.81024

[14] Dunford (N.) and Schwartz (J.T.).— Linear operators. Part II. Spectral theory. Self adjoint operators in Hilbert space. With the assistance of G. Bade and R.G. Bartle. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, (1988). | MR 1009163 | Zbl 0635.47002

[15] Fujiwara (K.).— Laplacians on rapidly branching trees, Duke Math. Jour., 83, No. 1, p. 191-202 (1996). | MR 1388848 | Zbl 0856.58044

[16] Golénia (S.).— Hardy inequality and Weyl asymptotic for discrete Laplacians, J. Funct. Anal. 266, no. 5, p. 2662-2688 (2014). | MR 3158705 | Zbl 1292.35300

[17] Haeseler (S.) and Keller (M.).— Generalized solutions and spectrum for Dirichlet forms on graphs, Boundaries and Spectral Theory, Progress in Probability, Birkhäuser, p. 181-201 (2011). | MR 3051699 | Zbl 1227.47023

[18] Keller (M.).— The essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346, Issue 1, p. 51-66 (2010). | MR 2558886 | Zbl 1285.05115

[19] Keller (M.) and Lenz (D.).— Unbounded Laplacians on graphs: Basic spectral properties and the heat equation, Math. Model. Nat. Phenom. Vol. 5, No. 2 (2009). | EuDML 197705 | MR 2662456 | Zbl 1207.47032

[20] Keller (M.) and Lenz (D.).— Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine Angew. Math. 666, p. 189-223 (2012). | MR 2920886 | Zbl 1252.47090

[21] Keller (M.), Lenz (D.), and Wojciechowski (R.).— Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs, Math. Z. 274, no. 3-4, p. 905-932 (2013). | MR 3078252 | Zbl 1269.05051

[22] Mohar (B.).— Isoperimetics inequalities, growth and the spectrum of graphs, Linear Algebra Appl. 103, p. 119-131 (1988). | MR 943998 | Zbl 0658.05055

[23] Mohar (B.).— Some relations between analytic and geometric properties of infinite graphs, Discrete Math. 95, no. 1-3, p. 193-219 (1991). | MR 1141939 | Zbl 0801.05051

[24] Milatovic (O.) and Truc (F.).— Self-adjoint extensions of discrete magnetic Schrödinger operators, Ann. Henri Poincaré 15, no. 5, p. 917-936 (2014). | MR 3192653 | Zbl 1288.81039

[25] Norris (J.).— Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics (1997). | MR 1600720 | Zbl 0938.60058

[26] Reed (M.) and Simon (B.).— Methods of Modern Mathematical Physics, Tome I-IV: Analysis of operators Academic Press. | Zbl 0459.46001

[27] Simon (B.).— Ergodic semigroups of positivity preserving self-adjoint operators. J. Functional Analysis 12, p. 335-339 (1973). | MR 358434 | Zbl 0252.47034

[28] Surchat (D.).— Infinité de valeurs propres sous le spectre essentiel du Laplacien d’un graphe, Phd Thesis (1993).

[29] Wojciechowski (R.).— Stochastic completeness of graphs, Ph.D. Thesis (2007), arXiv:0712.1570v2 | MR 2711706

[30] Wang (F.Y.).— Functional inequalities for empty essential spectrum, J. Funct. Anal. 170, no. 1, p. 219-245 (2000). | MR 1736202 | Zbl 0946.58010

[31] Wang (F.Y.).— Functional inequalities, semigroup properties and spectrum estimates, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3, no. 2, p. 263-295 (2000). | MR 1812701 | Zbl 1037.47505

[32] Wang (F.Y.).— Functional inequalities and spectrum estimates: the infinite measure case, J. Funct. Anal. 194, no. 2, p. 288-310 (2002). | MR 1934605 | Zbl 1021.58007