Neumann Domains on Quantum Graphs

Alon, Lior; Band, Ram

doi:10.48550/arxiv.1911.12435

Cited by 4 publications

(17 citation statements)

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“…Just as the basic idea behind Theorem 1.1 is gluing together nodal domains of Neumann eigenfunctions of the partition clusters to construct a test partition, here we will be interested in gluing together the so-called Neumann domains of the cluster Dirichlet eigenfunctions (see, e.g., [AlBa19,ABBE20]).…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…In fact, (1.2) can also be recovered from our proofs (see Remarks 4.2 and 5.3). Despite the completely different approaches (here we study cutting and pasting eigenfunctions arising from different minimal partitions, in [AlBa19] the point of departure being the whole graph eigenfunctions) this hints at a much deeper connection between these spectral minimal partitions and the nodal and Neumann domain patterns of the whole graph eigenfunctions, analogous to or extending the connection between nodal domains and partitions explored in [BBRS12], which will be left to future investigation to explore fully. Somewhat related is the idea of changing a vertex condition from standard to Dirichlet (or vice versa), another finite rank perturbation which leads to interlacing inequalities between Dirichlet and standard Laplacian eigenvalues.…”

Section: Introductionmentioning

confidence: 97%

“…The number and distribution of these has been explored at some length; see for example [ABB18,BBRS12,Ber08]. The Neumann domains arise as the clusters of a partition cut at the points where u k (x) = 0; the number of Neumann domains behaves similarly as a function of k, at least in the "generic" case where (among other things) all cuts are made away from the vertices [AlBa19]. Perhaps most notably for us, it has been shown in the generic case that the difference between the number of nodal domains ν(k) and the number of Neumann domains ξ(k) of u k satisfies exactly the same bounds as the indices appearing in Theorems 1.1 and 1.2 [AlBa19, Proposition 3.1(1)] (see also [ABBE20,Proposition 11.2]):…”

Section: Introductionmentioning

confidence: 99%

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Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Hofmann,

Kennedy

2021

Preprint

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We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [Kennedy et al, arXiv:2005.01126]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of cutting a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its rank. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander's inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Hofmann,

Kennedy

2021

Preprint

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“…Examples of functions having an oracle include the distance to the next nearest eigenvalue k n+1 [11], eigenfunction statistics [20,25], band widths for periodic graphs [7,30] as well 8 Even real analytic [4]. as the nodal surplus of f n [6,5] and Neumann surplus [4]. Surprisingly, 9 the oracle in the latter case does not dependend on .…”

Section: Secular Averagesmentioning

confidence: 99%

Universality of nodal count distribution in large metric graphs

Alon,

Band,

Berkolaiko

2021

Preprint

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An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph's first Betti number β. We study the distribution of the nodal surplus values in the countably infinite set of the graph's eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus. Contents1. Introduction 2. Definitions and preliminaries 3. Conjecture and main results 3.1. Analytical evidence in support of Conjecture 3.1 3.2. Efficient computation of the nodal surplus distribution 3.3. The polytope of nodal surplus distributions of a given graph 3.4. Numerical evidence in support of Conjecture 3.1 4. Secular averages 4.1. The secular manifold 4.2. The Barra-Gaspard method 4.3. BG method as a torus intergal 5. The nodal surplus distribution 5.1. The nodal surplus oracle function 5.2. Hessian in terms of the unitary evolution matrices 5.3. Proof of Theorem 3.7 and its generalization to graphs with loops. 6. The polytope of nodal surplus distributions and discrete graphs symmetries. 7. Mandarins and stowers -proof of Theorem 3.4

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“…Similar ideas may also be developed if more general conditions -say, δ-couplings -should imposed at the cut points, analogously to what was done in [BFG18] for the case of domains, although we will not develop such ideas here. Neumann domains, a Neumann-type analogue of nodal domains, have been studied recently on quantum graphs [AB19,ABBE18], and it is natural to ask whether there is a similar link between these and Neumann-type partitions as there is between Dirichlet partitions and nodal domains. We also leave this question to future work.…”

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confidence: 99%

A theory of spectral partitions of metric graphs

Kennedy¹,

Kurasov²,

Léna³

et al. 2020

Preprint

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We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in [Band et al, Comm. Math. Phys. 311 (2012), 815-838] as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic -rather than numerical -results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in [

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Neumann Domains on Quantum Graphs

Cited by 4 publications

References 29 publications

Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Universality of nodal count distribution in large metric graphs

A theory of spectral partitions of metric graphs

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