Sobolev-type inequalities and eigenvalue growth on graphs with finite measure

Hua, Bobo; Keller, Matthias; Schwarz, Michael; Wirth, Melchior

doi:10.1090/proc/14361

Sobolev-type inequalities and eigenvalue growth on graphs with finite measure
HTML articles powered by AMS MathViewer

by Bobo Hua, Matthias Keller, Michael Schwarz and Melchior Wirth;

Proc. Amer. Math. Soc. 151 (2023), 3401-3414

DOI: https://doi.org/10.1090/proc/14361

Published electronically: April 28, 2023

HTML | PDF | Request permission

Abstract:

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds cannot be established.

References

Michel Bonnefont, Sylvain Golénia, and Matthias Keller, Eigenvalue asymptotics for Schrödinger operators on sparse graphs, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 5, 1969–1998 (English, with English and French summaries). MR 3449203, DOI 10.5802/aif.2979
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289–297. MR 378001, DOI 10.1007/BF01214381
Shiu Yuen Cheng and Peter Li, Heat kernel estimates and lower bound of eigenvalues, Comment. Math. Helv. 56 (1981), no. 3, 327–338. MR 639355, DOI 10.1007/BF02566216
F. R. K. Chung and S.-T. Yau, Eigenvalues of graphs and Sobolev inequalities, Combin. Probab. Comput. 4 (1995), no. 1, 11–25. MR 1336652, DOI 10.1017/S0963548300001449
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, 1953. MR 65391
Harold Donnelly and Peter Li, Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J. 46 (1979), no. 3, 497–503. MR 544241
Agelos Georgakopoulos, Sebastian Haeseler, Matthias Keller, Daniel Lenz, and Radosław K. Wojciechowski, Graphs of finite measure, J. Math. Pures Appl. (9) 103 (2015), no. 5, 1093–1131 (English, with English and French summaries). MR 3333051, DOI 10.1016/j.matpur.2014.10.006
Bobo Hua and Delio Mugnolo, Time regularity and long-time behavior of parabolic $p$-Laplace equations on infinite graphs, J. Differential Equations 259 (2015), no. 11, 6162–6190. MR 3397320, DOI 10.1016/j.jde.2015.07.018
Matthias Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346 (2010), no. 1, 51–66. MR 2558886, DOI 10.1007/s00208-009-0384-y
M. Keller and D. Lenz, Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5 (2010), no. 4, 198–224. MR 2662456, DOI 10.1051/mmnp/20105409
Matthias Keller and Daniel Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine Angew. Math. 666 (2012), 189–223. MR 2920886, DOI 10.1515/CRELLE.2011.122
Matthias Keller, Daniel Lenz, Florentin Münch, Marcel Schmidt, and András Telcs, Note on short-time behavior of semigroups associated to self-adjoint operators, Bull. Lond. Math. Soc. 48 (2016), no. 6, 935–944. MR 3608938, DOI 10.1112/blms/bdw054
Matthias Keller, Daniel Lenz, Marcel Schmidt, and Radosław K. Wojciechowski, Note on uniformly transient graphs, Rev. Mat. Iberoam. 33 (2017), no. 3, 831–860. MR 3713033, DOI 10.4171/RMI/957
Matthias Keller, Shiping Liu, and Peyerimhoff Norbert, A note on eigenvalue bounds for non-compact manifolds, Preprint arXiv:1706.02437 (2017).
Matthias Keller and Michael Schwarz, The Kazdan-Warner equation on canonically compactifiable graphs, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 70, 18. MR 3776360, DOI 10.1007/s00526-018-1329-7
Daniel Lenz, Marcel Schmidt, and Peter Stollmann, Topological Poincaré type inequalities and lower bounds on the infimum of the spectrum for graphs, ArXiv e-prints (2018).
G. V. Rozenblyum and M. Z. Solomyak, On spectral estimates for Schrödinger-type operators: the case of small local dimension, Funktsional. Anal. i Prilozhen. 44 (2010), no. 4, 21–33 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 44 (2010), no. 4, 259–269. MR 2768562, DOI 10.1007/s10688-010-0037-7
Buchin Su, Lectures on differential geometry, World Scientific Publishing Co., Singapore, 1980. Translated from the Chinese by K. T. Chang; With a foreword by S. S. Chern. MR 664600
Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR 1743100, DOI 10.1017/CBO9780511470967
Radoslaw Krzysztof Wojciechowski, Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–City University of New York. MR 2711706