On the hot spots of quantum graphs

James B. Kennedy; Jonathan Rohleder

doi:10.3934/cpaa.2021095

Article Contents

2021, Volume 20, Issue 9: 3029-3063. Doi: 10.3934/cpaa.2021095

This issue Previous Article Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay Next Article Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions

On the hot spots of quantum graphs

James B. Kennedy^1, and
Jonathan Rohleder^2, ,

1.
Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa, Portugal
2.
Matematiska institutionen, Stockholms universitet, 106 91 Stockholm, Sweden

^* Corresponding author
^* Corresponding author

Received: October 2020

Revised: April 2021

Early access: May 2021

Published: September 2021

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Abstract

We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity–Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.

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Mathematics Subject Classification: Primary: 34B45; Secondary: 34L10, 35R02, 81Q35.

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Figure 3.1. Left: a path graph with the hot spots marked in grey. Right: a cycle graph; here every point is a hot spot

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Figure 3.2. A pumpkin and a star graph. The set $ M (\Gamma) $ in the equilateral case is marked in grey

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Figure 3.3. A flower graph and a complete graph. The hot spots for the equilateral case are marked in grey

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Figure 3.4. A lasso graph with its hot spots in grey

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Figure 4.1. The perturbed "figure-8" graph of Example 4.7 and its hot spots in grey

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Figure 4.2. The perturbed path graph of Example 4.8 and its hot spots in grey

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Figure 5.1. The complete graph admits an eigenfunction for $ \mu_2 $ whose maximum is at $ v_0 $ and minimum is achieved at the other $ v_k $, with no other critical points

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Figure 6.1. The star graph $ \Gamma^* $, the intermediate tree $ \Gamma^\top $, and the final, symmetric tree $ \Gamma $ in the case $ m = 5 $

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Figure 7.1. The discrete graph $ {{\mathcal G}} $, a "pumpkin on a stick"

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Figure 7.2. Metric graph incarnations of the discrete graph $ {{\mathcal G}} $ from Figure 7.1 together with their hot spots (grey)

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Figure 8.1. A graph for which we expect that $ M = \{v_-, v_2, v_3\} $ is possible even if the edge lengths are incommensurable

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Figure 8.2. A candidate graph for Conjecture 8.9. Candidate locations for the hot spots are marked in grey (the precise location will depend on the respective edge lengths and the "thickness" of the pumpkins, and could be within the pumpkins)

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References

[1]	R. Adami, E. Serra and P. Tilli, Negative energy ground states for the $L^2$-critical NLSE on metric graphs, Commun. Math. Phys., 352 (2017), 387-406. doi: 10.1007/s00220-016-2797-2.
[2]	R. Adami, E. Serra and P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal., 271 (2016), 201-223. doi: 10.1016/j.jfa.2016.04.004.
[3]	R. Adami, E. Serra and P. Tilli, NLS ground states on graphs, Calc. Var. Partial Differ. Equ., 54 (2015), 743-761. doi: 10.1007/s00526-014-0804-z.
[4]	M. Aizenman, H. Schanz, U. Smilansky and S. Warzel, Edge switching transformations of quantum graphs, Acta Phys. Polon. A, 132 (2017), 1699-1703.
[5]	L. Alon, Quantum graphs–Generic eigenfunctions and their nodal count and Neumann count statistics, Ph.D thesis, Technion, Israel, arXiv: 2010.03004.
[6]	L. Alon and R. Band, Neumann domains on quantum graphs, arXiv: 1911.12435.
[7]	L. Alon, R. Band and G. Berkolaiko, Nodal statistics on quantum graphs, Commun. Math. Phys., 362 (2018), 909-948. doi: 10.1007/s00220-018-3111-2.
[8]	L. Alon, R. Band, M. Bersudsky and S. Egger, Neumann Domains on Graphs and Manifolds, arXiv: 1805.07612.
[9]	S Ariturk, Eigenvalue estimates on quantum graphs, arXiv: 1609.07471.
[10]	R. Band, The nodal count $\{0, 1, 2, 3, \ldots\}$ implies the graph is a tree, Philos. Trans. R. Soc. Lond. A, 372 (2014), (24pp). doi: 10.1098/rsta.2012.0504.
[11]	R. Band, G. Berkolaiko, H. Raz and U. Smilansky, The number of nodal domains of graphs as a stability index of graph partitions, Commun. Math. Phys., 311 (2012), 815-838. doi: 10.1007/s00220-011-1384-9.
[12]	R. Band, G. Berkolaiko and U. Smilansky, Dynamics of nodal points and the nodal count of a family of quantum graphs, Ann. Henri Poincaré, 13 (2012), 145-184. doi: 10.1007/s00023-011-0124-1.
[13]	R. Band and D. Fajman, Topological properties of Neumann domains, Ann. Henri Poincaré, 17 (2016), 2379-2407. doi: 10.1007/s00023-016-0468-7.
[14]	R. Band and G. Lévy, Quantum graphs which optimize the spectral gap, Ann. Henri Poincaré, 18 (2017), 3269-3323. doi: 10.1007/s00023-017-0601-2.
[15]	R. Bañuelos and K. Burdzy, On the "hot spots" conjecture of J. Rauch, J. Funct. Anal., 164 (1999), 1-33. doi: 10.1006/jfan.1999.3397.
[16]	G. Berkolaiko, J. B. Kennedy, P. Kurasov and D. Mugnolo, Surgery principles for the spectral analysis of quantum graphs, Trans. Amer. Math. Soc., 372 (2019), 5153-5197. doi: 10.1090/tran/7864.
[17]	G. Berkolaiko, J. B. Kennedy, P. Kurasov and D. Mugnolo, Edge connectivity and the spectral gap of combinatorial and quantum graphs, J. Phys. A: Math. Theor., 50 (2017), 365201 (29pp). doi: 10.1088/1751-8121/aa8125.
[18]	G. Berkolaiko and P. Kuchment, Introduction to quantum graphs. Math. Surveys and Monographs vol. 186, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.
[19]	G. Berkolaiko and P. Kuchment, Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths, Spectral Geometry, 117–137, Proc. Sympos. Pure Math., vol.84, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/pspum/084/1352.
[20]	G. Berkolaiko, Y. Latushkin and S. Sukhtaiev, Limits of quantum graph operators with shrinking edges, Adv. Math., 352 (2019), 632-669. doi: 10.1016/j.aim.2019.06.017.
[21]	G. Berkolaiko and W. Liu, Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph, J. Math. Anal. Appl., 445 (2017), 803-818. doi: 10.1016/j.jmaa.2016.07.026.
[22]	D. Borthwick, L. Corsi and K. Jones, Sharp diameter bound on the spectral gap for quantum graphs, arXiv: 1905.03071.
[23]	K. Burdzy and W. Werner, A counterexample to the "hot spots" conjecture, Ann. Math., 149 (1999), 309-317. doi: 10.2307/121027.
[24]	C. Cacciapuoti, Scale invariant effective Hamiltonians for a graph with a small compact core, Symmetry, 11 (2019), 359.
[25]	C. Cacciapuoti, D. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30 (2017), 3271-3303. doi: 10.1088/1361-6544/aa7cc3.
[26]	J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian. in Problems in analysis, Princeton Univ. Press, Princeton, N. J., 1970, 195-199.
[27]	M. K. Chung, S. Seo, N. Adluru and H. K. Vorperian, Hot Spots Conjecture and Its Application to Modeling Tubular Structures. In K. Suzuki, F. Wang, D. Shen and P. Yan (eds), Machine Learning in Medical Imaging, Lecture Notes in Computer Science, vol. 7009, Springer, Berlin–Heidelberg, 2011,225–232.
[28]	S. Dovetta and L. Tentarelli, $L^2$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features, Calc. Var. Partial Differ. Equ., 58 (2019), 26 pp. doi: 10.1007/s00526-019-1565-5.
[29]	L. C. Evans, The Fiedler Rose: On the extreme points of the Fiedler vector, arXiv: 1112.6323.
[30]	L. Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier (Grenoble), 55 (2005), 199-211.
[31]	L. Friedlander, Genericity of simple eigenvalues for a metric graph, Israel J. Math., 146 (2005), 149-156. doi: 10.1007/BF02773531.
[32]	H. Gernandt and J. P. Pade, Schur reduction of trees and extremal entries of the Fiedler vector, Linear Algebra Appl., 570 (2019), 93-122. doi: 10.1016/j.laa.2019.02.008.
[33]	S. Gnutzmann, U. Smilansky and J. Weber, Nodal counting on quantum graphs, Special section on quantum graphs, Waves Random Media, 14 (2004), S61–S73. doi: 10.1088/0959-7174/14/1/011.
[34]	E. M. Harrell II and A. V. Maltsev, Localization and landscape functions on quantum graphs, arXiv: 1803.01186. doi: 10.1090/tran/7908.
[35]	E. M. Harrell II and A. V. Maltsev, On Agmon metrics and exponential localization for quantum graphs, Commun. Math. Phys., 359 (2018), 429-448. doi: 10.1007/s00220-018-3124-x.
[36]	M. Hofmann, An existence theory for nonlinear equations on metric graphs via energy methods, arXiv: 1909.07856.
[37]	M. Hofmann, J. B. Kennedy, D. Mugnolo and M. Plümer, Asymptotics and estimates for spectral minimal partitions of metric graphs, arXiv: 2007.01412.
[38]	C. Judge and S. Mondal, Euclidean triangles have no hot spots, Ann. Math., 191 (2020), 167-211. doi: 10.4007/annals.2020.191.1.3.
[39]	A. Kairzhan, D. E. Pelinovsky and R. H. Goodman, Drift of spectrally stable shifted states on star graphs, SIAM J. Appl. Dyn. Syst., 18 (2019), 1723–1755. doi: 10.1137/19M1246146.
[40]	G. Karreskog, P. Kurasov and I. Trygg Kupersmidt, Schrödinger operators on graphs: symmetrization and Eulerian cycles, Proc. Amer. Math. Soc., 144 (2016) 1197–1207. doi: 10.1090/proc12784.
[41]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.
[42]	J. B. Kennedy, P. Kurasov, C. Léna and D. Mugnolo, A theory of spectral partitions of metric graphs, Calc. Var. Partial Differ. Equ., 60 (2021), 63 pp. doi: 10.1007/s00526-021-01966-y.
[43]	J. B. Kennedy, P. Kurasov, G. Malenová and D. Mugnolo, On the spectral gap of a quantum graph, Ann. Henri Poincaré, 17 (2016), 2439-2473. doi: 10.1007/s00023-016-0460-2.
[44]	J. B. Kennedy and J. Rohleder, On the hot spots of quantum trees, Proc. Appl. Math. Mech., 18 (2018), e201800122.
[45]	D. Krejčiřík and M. Tušek, Location of hot spots in thin curved strips, J. Differ. Equ., 266 (2019), 2953-2977. doi: 10.1016/j.jde.2018.08.053.
[46]	P. Kurasov, G. Malenová and S. Naboko, Spectral gap for quantum graphs and their connectivity, J. Phys. A, 46 (2013), 275309. doi: 10.1088/1751-8113/46/27/275309.
[47]	P. Kurasov and S. Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory, 4 (2014), 211-219. doi: 10.4171/JST/67.
[48]	C. Lange, S. Liu, N. Peyerimhoff and O. Post, Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians, Calc. Var. Partial Differ. Equ., 54 (2015), 4165-4196. doi: 10.1007/s00526-015-0935-x.
[49]	R. Lederman and S. Steinerberger, Extreme values of the Fiedler vector on trees, arXiv: 1912.08327.
[50]	J. R. Lee, S. O. Gharan and L. Trevisan, Multiway spectral partitioning and higher-order Cheeger inequalities, J. ACM, 61 (2014), 30 pp. doi: 10.1145/2665063.
[51]	J. Lefèvre, Fiedler vectors and elongation of graphs: a threshold phenomenon on a particular class of trees, arXiv: 1302.1266.
[52]	J. Rohleder, Eigenvalue estimates for the Laplacian on a metric tree, Proc. Amer. Math. Soc., 145 (2017), 2119-2129. doi: 10.1090/proc/13403.
[53]	J. Rohleder and C. Seifert, Spectral monotonicity for Schrödinger operators on metric graphs, Oper. Theory Adv. Appl., 281 (2020), 291-310.
[54]	B. Siudeja, Hot spots conjecture for a class of acute triangles, Math. Z., 280 (2015), 783-806. doi: 10.1007/s00209-015-1448-1.
[55]	S. Steinerberger, Hot Spots in Convex Domains are in the Tips (up to an Inradius), Commun. Partial Differ. Equ., 45 (2020), 641-654. doi: 10.1080/03605302.2020.1750427.