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Surgery principles for the spectral analysis of quantum graphs
Authors: Gregory Berkolaiko, James B. Kennedy, Pavel Kurasov and Delio Mugnolo
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 34B45; Secondary 05C50, 35P15, 81Q35
DOI: https://doi.org/10.1090/tran/7864
Published electronically: June 21, 2019
Abstract | References | Similar Articles | Additional Information
Abstract: We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet, or -type) which show how various types of changes of a local or localised nature to a graph impact on the spectrum of the Laplacian. Many of these principles are entirely new; these include ``transplantation'' of volume within a graph based on the behaviour of its eigenfunctions, as well as ``unfolding'' of local cycles and pendants. In other cases we establish sharp generalisations, extensions, and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions, and introducing new pendant subgraphs.
To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a compact metric graph to estimate the lowest non-trivial eigenvalue of the Laplacian with natural vertex conditions. This quantitative isoperimetric-type inequality interpolates between two known estimates -- one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound) -- and includes them as special cases.
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Riccardo Adami, Enrico Serra, and Paolo Tilli, Negative energy ground states for the -critical NLSE on metric graphs, Comm. Math. Phys. 352 (2017), no. 1, 387-406. MR 3623262, https://doi.org/10.1007/s00220-016-2797-2
Riccardo Adami, Enrico Serra, and Paolo Tilli, Lack of ground state for NLSE on bridge-type graphs, Mathematical technology of networks, Springer Proc. Math. Stat., vol. 128, Springer, Cham, 2015, pp. 1-11. MR 3375151, https://doi.org/10.1007/978-3-319-16619-3_1
Riccardo Adami, Enrico Serra, and Paolo Tilli, NLS ground states on graphs, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 743-761. MR 3385179, https://doi.org/10.1007/s00526-014-0804-z
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M. Aizenman, H. Schanz, U. Smilansky, and S. Warzel, Edge switching transformations of quantum graphs, Acta Phys. Polon. A 132 (2017), 1699-1703.
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S. Ariturk, Eigenvalue estimates on quantum graphs, preprint, arXiv:1609.07471, 2016.
Ram Band and Guillaume Lévy, Quantum graphs which optimize the spectral gap, Ann. Henri Poincaré 18 (2017), no. 10, 3269-3323. MR 3697195, https://doi.org/10.1007/s00023-017-0601-2
[BBJL17]
R. Band, G. Berkolaiko, C. H. Joyner, and W. Liu, Quotients of finite-dimensional operators by symmetry representations, preprint, arXiv:1711.00918, 2017.
Ram Band, Gregory Berkolaiko, Hillel Raz, and Uzy Smilansky, The number of nodal domains on quantum graphs as a stability index of graph partitions, Comm. Math. Phys. 311 (2012), no. 3, 815-838. MR 2909765, https://doi.org/10.1007/s00220-011-1384-9
Joachim von Below, A characteristic equation associated to an eigenvalue problem on -networks, Linear Algebra Appl. 71 (1985), 309-325. MR 813056, https://doi.org/10.1016/0024-3795(85)90258-7
Gregory Berkolaiko, A lower bound for nodal count on discrete and metric graphs, Comm. Math. Phys. 278 (2008), no. 3, 803-819. MR 2373444, https://doi.org/10.1007/s00220-007-0391-3
Gregory Berkolaiko, An elementary introduction to quantum graphs, Geometric and computational spectral theory, Contemp. Math., vol. 700, Amer. Math. Soc., Providence, RI, 2017, pp. 41-72. MR 3748521, https://doi.org/10.1090/conm/700/14182
G. Berkolaiko, E. B. Bogomolny, and J. P. Keating, Star graphs and Šeba billiards, J. Phys. A 34 (2001), no. 3, 335-350. MR 1822781, https://doi.org/10.1088/0305-4470/34/3/301
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, https://doi.org/10.1090/pspum/084/1352
[BLS18]
G. Berkolaiko, Yu. Latushkin, and S. Sukhtaiev, Limits of quantum graph operators with shrinking edges, preprint, arXiv:1806.00561, 2018.
Gregory Berkolaiko and Wen Liu, Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph, J. Math. Anal. Appl. 445 (2017), no. 1, 803-818. MR 3543796, https://doi.org/10.1016/j.jmaa.2016.07.026
A. N. Bondarenko and V. A. Dedok, The technique of spectral surgery for quantum graphs, Dokl. Akad. Nauk 444 (2012), no. 5, 473-476 (Russian); English transl., Dokl. Math. 85 (2012), no. 3, 384-387. MR 3027128, https://doi.org/10.1134/S106456241203026X
Lorenzo Brasco and Guido De Philippis, Spectral inequalities in quantitative form, Shape optimization and spectral theory, De Gruyter Open, Warsaw, 2017, pp. 201-281. MR 3681151
Yves Colin de Verdière, Semi-classical measures on quantum graphs and the Gauß map of the determinant manifold, Ann. Henri Poincaré 16 (2015), no. 2, 347-364. MR 3302601, https://doi.org/10.1007/s00023-014-0326-4
[Eva11]
L. C. Evans, The Fiedler rose: On the extreme points of the Fiedler vector, preprint, arXiv:1112.6323, 2011.
Pavel Exner and Michal Jex, On the ground state of quantum graphs with attractive -coupling, Phys. Lett. A 376 (2012), no. 5, 713-717. MR 2880105, https://doi.org/10.1016/j.physleta.2011.12.035
Shaun Fallat and Steve Kirkland, Extremizing algebraic connectivity subject to graph-theoretic constraints, Electron. J. Linear Algebra 3 (1998), 48-74. MR 1624787, https://doi.org/10.13001/1081-3810.1014
Miroslav Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23(98) (1973), 298-305. MR 0318007
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
Leonid Friedlander, Genericity of simple eigenvalues for a metric graph, Israel J. Math. 146 (2005), 149-156. MR 2151598, https://doi.org/10.1007/BF02773531
Ji-Ming Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008), no. 23, 5702-5711. MR 2459389, https://doi.org/10.1016/j.disc.2007.10.044
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, https://doi.org/10.1090/proc12784
Tosio Kato, Perturbation theory for linear operators, 2nd ed., with Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
[Ken18]
J. B. Kennedy, A sharp eigenvalue bound for quantum graphs in terms of their diameter, preprint, arXiv:1807.08185, 2018.
[KKLM19]
J. B. Kennedy, P. Kurasov, C. Léna and D. Mugnolo, A theory of spectral partitions of metric graphs, in preparation, 2019.
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, https://doi.org/10.1007/s00023-016-0460-2
[KR18]
J. Kennedy and J. Rohleder, On the hot spots of quantum trees, Proc. Appl. Math. Mech., 18 (2018), e201800122.
Pavel Kurasov and Sergey Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory 4 (2014), no. 2, 211-219. MR 3232809, https://doi.org/10.4171/JST/67
Ulrike von Luxburg, A tutorial on spectral clustering, Stat. Comput. 17 (2007), no. 4, 395-416. MR 2409803, https://doi.org/10.1007/s11222-007-9033-z
Jeremy L. Marzuola and Dmitry E. Pelinovsky, Ground state on the dumbbell graph, Appl. Math. Res. Express. AMRX 1 (2016), 98-145. MR 3483843, https://doi.org/10.1093/amrx/abv011
Serge Nicaise, Spectre des réseaux topologiques finis, Bull. Sci. Math. (2) 111 (1987), no. 4, 401-413 (French, with English summary). MR 921561
Jonathan Rohleder, Eigenvalue estimates for the Laplacian on a metric tree, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2119-2129. MR 3611325, https://doi.org/10.1090/proc/13403
[RoSe18]
J. Rohleder and C. Seifert, Spectral monotonicity for Schrödinger operators on metric graphs, preprint, arXiv:1804.01827, 2018.
Michael Solomyak, On eigenvalue estimates for the weighted Laplacian on metric graphs, Nonlinear problems in mathematical physics and related topics, I, Int. Math. Ser. (N. Y.), vol. 1, Kluwer/Plenum, New York, 2002, pp. 327-347. MR 1970620, https://doi.org/10.1007/978-1-4615-0777-2_20
Alexander Weinstein and William Stenger, Methods of intermediate problems for eigenvalues, with Theory and ramifications, Mathematics in Science and Engineering, Vol. 89, Academic Press, New York-London, 1972. MR 0477971
[XLS18]
J. Xue, H. Lin, and J. Shu, The algebraic connectivity of graphs with given circumference, Theoret. Comput. Sci., in press, 2018.
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 34B45, 05C50, 35P15, 81Q35
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Gregory Berkolaiko
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: gregory.berkolaiko@math.tamu.edu
James B. Kennedy
Affiliation: Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, P-1749-016 Lisboa, Portugal
Email: jbkennedy@fc.ul.pt
Pavel Kurasov
Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
Email: kurasov@math.su.se
Delio Mugnolo
Affiliation: Lehrgebiet Analysis, Fakultät Mathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany
Email: delio.mugnolo@fernuni-hagen.de
Keywords: Quantum graphs, spectral geometry of quantum graphs, bounds on spectral gaps
Received by editor(s): August 3, 2018
Received by editor(s) in revised form: March 11, 2019
Additional Notes: The work of the first author was partially supported by the NSF under grant DMS-1410657.
The second author was supported by the Fundação para a Ciência e a Tecnologia, Portugal, via the program "Investigador FCT", reference IF/01461/2015, and project PTDC/MAT-CAL/4334/2014.
The third author was partially supported by the Swedish Research Council (Grant D0497301).
The fourth author was partially supported by the Deutsche Forschungsgemeinschaft (Grant 397230547).
All four authors were partially supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld, Germany, within the framework of the cooperation group on "Discrete and continuous models in the theory of networks".
Article copyright: © Copyright 2019 American Mathematical Society
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