Surgery principles for the spectral analysis of quantum graphs

Berkolaiko, Gregory; Kennedy, James; Kurasov, Pavel; Mugnolo, Delio

doi:10.1090/tran/7864

Surgery principles for the spectral analysis of quantum graphs
HTML articles powered by AMS MathViewer

by Gregory Berkolaiko, James B. Kennedy, Pavel Kurasov and Delio Mugnolo PDF

Trans. Amer. Math. Soc. 372 (2019), 5153-5197 Request permission

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 $\delta$-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.

References

R. Adami, Ground states for NLS on graphs: a subtle interplay of metric and topology, Math. Model. Nat. Phenom. 11 (2016), no. 2, 20–35. MR 3491787, DOI 10.1051/mmnp/201611202
Riccardo Adami, Enrico Serra, and Paolo Tilli, Negative energy ground states for the $L^2$-critical NLSE on metric graphs, Comm. Math. Phys. 352 (2017), no. 1, 387–406. MR 3623262, DOI 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, DOI 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, DOI 10.1007/s00526-014-0804-z
M. Aizenman, H. Schanz, U. Smilansky, and S. Warzel, Edge switching transformations of quantum graphs, Acta Phys. Polon. A 132 (2017), 1699–1703.
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, DOI 10.1007/s00023-017-0601-2
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, DOI 10.1007/s00220-011-1384-9
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
Joachim von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks, Linear Algebra Appl. 71 (1985), 309–325. MR 813056, DOI 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, DOI 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, DOI 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, DOI 10.1088/0305-4470/34/3/301
Gregory Berkolaiko, James B. Kennedy, Pavel Kurasov, and Delio Mugnolo, Edge connectivity and the spectral gap of combinatorial and quantum graphs, J. Phys. A 50 (2017), no. 36, 365201, 29. MR 3688110, DOI 10.1088/1751-8121/aa8125
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
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, DOI 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, DOI 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, DOI 10.1515/9783110550887-007
Yves Colin de Verdière, Semi-classical measures on quantum graphs and the Gaußmap of the determinant manifold, Ann. Henri Poincaré 16 (2015), no. 2, 347–364. MR 3302601, DOI 10.1007/s00023-014-0326-4
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
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 $\delta$-coupling, Phys. Lett. A 376 (2012), no. 5, 713–717. MR 2880105, DOI 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, DOI 10.13001/1081-3810.1014
Miroslav Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23(98) (1973), 298–305. MR 318007
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, DOI 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, DOI 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, DOI 10.1090/proc12784
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
J. B. Kennedy, A sharp eigenvalue bound for quantum graphs in terms of their diameter, preprint, arXiv:1807.08185, 2018.
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, DOI 10.1007/s00023-016-0460-2
J. Kennedy and J. Rohleder, On the hot spots of quantum trees, Proc. Appl. Math. Mech., 18 (2018), e201800122.
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
Ulrike von Luxburg, A tutorial on spectral clustering, Stat. Comput. 17 (2007), no. 4, 395–416. MR 2409803, DOI 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, DOI 10.1093/amrx/abv011
Delio Mugnolo, Semigroup methods for evolution equations on networks, Understanding Complex Systems, Springer, Cham, 2014. MR 3243602, DOI 10.1007/978-3-319-04621-1
Serge Nicaise, Spectre des réseaux topologiques finis, Bull. Sci. Math. (2) 111 (1987), no. 4, 401–413 (French, with English summary). MR 921561
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Jonathan Rohleder, Eigenvalue estimates for the Laplacian on a metric tree, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2119–2129. MR 3611325, DOI 10.1090/proc/13403
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, DOI 10.1007/978-1-4615-0777-2_{2}0
Alexander Weinstein and William Stenger, Methods of intermediate problems for eigenvalues, Mathematics in Science and Engineering, Vol. 89, Academic Press, New York-London, 1972. Theory and ramifications. MR 0477971
J. Xue, H. Lin, and J. Shu, The algebraic connectivity of graphs with given circumference, Theoret. Comput. Sci., in press, 2018.