Nodal statistics on quantum graphs
type: Item
instance of: scholarly article
title: Nodal statistics on quantum graphs
zbMATH Open document ID: 1406.81037
author: Lior Alon
author: R. Band
author: Gregory Berkolaiko
published in: Communications in Mathematical Physics
publication date: 2018-09-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1709.10413
review text: The number \(\phi_n\) of zeroes of the \(n\)-th eigenfunction of the Laplacian with standard (Neumann-Kirchhoff) vertex conditions on a metric graph is (for an eigenfunction which does not vanish on any vertex) larger than, or equal to, \(n - 1\), with equality \(\phi_n = n - 1\) for all \(n\) if and only if the graph is a tree; this can be considered a generalization of Sturm's oscillation theorem. In this paper the authors study the \textit{nodal surplus} \[ \sigma_n := \phi_n - (n - 1) \] for general finite metric graphs with \(\beta\) cycles. They focus on the \textit{nodal surplus distribution} of a metric graph. In the first main result it is shown that the latter is a well-defined probability distribution on the set \(\{0, 1, \dots, \beta \}\) and that this distribution is symmetric; as a consequence of this, the Betti number \(\beta\) of the graph can be recovered as twice the average nodal surplus. The second main result deals with the class of graphs whose cycles are disjoint. In this situation it is shown that the nodal surplus distribution is Binomial with parameters \(\beta\) and \(1/2\).
reviewed by: Jonathan Rohleder
Mathematics Subject Classification ID: 81Q35
Mathematics Subject Classification ID: 34B45
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 05C12
zbMATH DE Number: 6936013
zbMATH Keywords: quantum graph
zbMATH Keywords: eigenfunction
zbMATH Keywords: nodal set
zbMATH Keywords: nodal statistics
MaRDI profile type: Publication
OpenAlex ID: W3105575315
arXiv ID: 1709.10413
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Wikidata QID: Q130161233
DOI: 10.1007/S00220-018-3111-2
description: scientific article