Eigenvalue estimates for the Laplacian on a metric tree

Rohleder, Jonathan

doi:10.1090/proc/13403

Eigenvalue estimates for the Laplacian on a metric tree
HTML articles powered by AMS MathViewer

by Jonathan Rohleder PDF

Proc. Amer. Math. Soc. 145 (2017), 2119-2129 Request permission

Abstract:

We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular, we establish a sharp upper bound for the spectral gap, i.e., the smallest positive eigenvalue, and show that equilateral star graphs are the unique maximizers of the spectral gap among all trees of a given average length.

References

Ram Band, Gregory Berkolaiko, and Tracy Weyand, Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs, J. Math. Phys. 56 (2015), no. 12, 122111, 20. MR 3434868, DOI 10.1063/1.4937119
G. Berkolaiko and P. Kuchment, Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths, Spectral geometry, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp. 117–137. MR 2985312, DOI 10.1090/pspum/084/1352
Gregory Berkolaiko and Peter Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. MR 3013208, DOI 10.1090/surv/186
L. M. Del Pezzo and J. D. Rossi, The first eigenvalue of the $p$-Laplacian on quantum graphs, Anal. Math. Phys., in press.
Semra Demirel-Frank, Spectral inequalities for quantum graphs, Mathematical technology of networks, Springer Proc. Math. Stat., vol. 128, Springer, Cham, 2015, pp. 65–80. MR 3375156, DOI 10.1007/978-3-319-16619-3_{6}
Semra Demirel and Evans M. Harrell II, On semiclassical and universal inequalities for eigenvalues of quantum graphs, Rev. Math. Phys. 22 (2010), no. 3, 305–329. MR 2647038, DOI 10.1142/S0129055X10003965
Tomas Ekholm, Rupert L. Frank, and Hynek Kovařík, Eigenvalue estimates for Schrödinger operators on metric trees, Adv. Math. 226 (2011), no. 6, 5165–5197. MR 2775897, DOI 10.1016/j.aim.2011.01.001
Pavel Exner and Michal Jex, On the ground state of quantum graphs with attractive $\delta$-coupling, Phys. Lett. A 376 (2012), no. 5, 713–717. MR 2880105, DOI 10.1016/j.physleta.2011.12.035
N. Filonov, On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator, Algebra i Analiz 16 (2004), no. 2, 172–176 (Russian); English transl., St. Petersburg Math. J. 16 (2005), no. 2, 413–416. MR 2068346, DOI 10.1090/S1061-0022-05-00857-5
Leonid Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal. 116 (1991), no. 2, 153–160. MR 1143438, DOI 10.1007/BF00375590
Leonid Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 199–211 (English, with English and French summaries). MR 2141695
G. Karreskog, P. Kurasov, and I. Trygg Kupersmidt, Schrödinger operators on graphs: symmetrization and Eulerian cycles, Proc. Amer. Math. Soc. 144 (2016), no. 3, 1197–1207. MR 3447672, DOI 10.1090/proc12784
James B. Kennedy, Pavel Kurasov, Gabriela Malenová, and Delio Mugnolo, On the spectral gap of a quantum graph, Ann. Henri Poincaré 17 (2016), no. 9, 2439–2473. MR 3535868, DOI 10.1007/s00023-016-0460-2
Pavel Kurasov, Spectral gap for complete graphs: upper and lower estimates, Mathematical technology of networks, Springer Proc. Math. Stat., vol. 128, Springer, Cham, 2015, pp. 121–132. MR 3375158, DOI 10.1007/978-3-319-16619-3_{8}
P. Kurasov, G. Malenová, and S. Naboko, Spectral gap for quantum graphs and their edge connectivity, J. Phys. A 46 (2013), no. 27, 275309, 16. MR 3081922, DOI 10.1088/1751-8113/46/27/275309
Pavel Kurasov and Sergey Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory 4 (2014), no. 2, 211–219. MR 3232809, DOI 10.4171/JST/67
Fernando Lledó and Olaf Post, Eigenvalue bracketing for discrete and metric graphs, J. Math. Anal. Appl. 348 (2008), no. 2, 806–833. MR 2446037, DOI 10.1016/j.jmaa.2008.07.029
K. Naimark and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proc. London Math. Soc. (3) 80 (2000), no. 3, 690–724. MR 1744781, DOI 10.1112/S0024611500012272
Serge Nicaise, Spectre des réseaux topologiques finis, Bull. Sci. Math. (2) 111 (1987), no. 4, 401–413 (French, with English summary). MR 921561