Eigenvalue and nodal properties on quantum graph trees*

Schapotschnikow, Philipp

doi:10.1080/17455030600702535

Cited by 28 publications

(46 citation statements)

References 6 publications

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“…This result for the interval is the famous Sturm oscillation theorem [1,2] and its generalization for trees was done in [43,44]. The fact that discrete tree graphs also have this nodal count is proved in [45][46][47].…”

Section: Definition 12mentioning

confidence: 96%

“…Observe that Γ has infinitely many generic eigenvalues as a direct conclusion of lemma 4.1. If Γ is a tree graph then it was proved in [43,44] (see also appendix A in [45]) that the nodal counts of all generic eigenfunctions are φ n = n − 1 and ν n = n. Otherwise, if Γ has β > 0 cycles, assume by contradiction that there are only finitely many generic eigenfunctions with φ n = n − 1. In particular, this means that there is at least one generic eigenfunction for which φ n = n − 1, and thus σ n = 0.…”

Section: Proofs For Metric Graphsmentioning

confidence: 99%

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The nodal count {0,1,2,3,…} implies the graph is a tree

Band

2014

Phil. Trans. R. Soc. A.

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Sturm's oscillation theorem states that the n th eigenfunction of a Sturm–Liouville operator on the interval has n −1 zeros (nodes) (Sturm 1836 J. Math. Pures Appl. 1 , 106–186; 373–444). This result was generalized for all metric tree graphs (Pokornyĭ et al. 1996 Mat. Zametki 60 , 468–470 ( doi:10.1007/BF02320380 ); Schapotschnikow 2006 Waves Random Complex Media 16 , 167–178 ( doi:10.1080/1745530600702535 )) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys. 278 , 803–819 ( doi:10.1007/S00220-007-0391-3 ); Dhar & Ramaswamy 1985 Phys. Rev. Lett. 54 , 1346–1349 ( doi:10.1103/PhysRevLett.54.1346 ); Fiedler 1975 Czechoslovak Math. J. 25 , 607–618). We prove the converse theorems for both discrete and metric graphs. Namely if for all n , the n th eigenfunction of the graph has n −1 zeros, then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues (Berkolaiko 2013 Anal. PDE 6 , 1213–1233 ( doi:10.2140/apde.2013.6.1213 ); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A 372 , 20120522 ( doi:10.1098/rsta.2012.0522 ); Colin de Verdière 2013 Anal. PDE 6 , 1235–1242 ( doi:10.2140/apde.2013.6.1235 )). In the course of the proof, we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of ‘discretized’ versions of a metric graph and prove that their nodal counts are related to those of the metric graph.

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Section: Definition 12mentioning

confidence: 96%

Section: Proofs For Metric Graphsmentioning

confidence: 99%

The nodal count {0,1,2,3,…} implies the graph is a tree

Band

2014

Phil. Trans. R. Soc. A.

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“…Then, by making use of Schapotschnikow's theorem [23], the metric nodal counts ν n = n. This applies in particular to the isospectral graphs of the types 7 3 , 13 3 and 13 8 which are tree graphs and…”

Section: The Metric Nodal Sequencesmentioning

confidence: 99%

“…Recently Schapotschnikow [23] proved that Sturm's Oscillation Theorem extends to finite tree (loop-less) graphs: the number of nodal domains of the n'th eigenfunction (ordered by increasing eigenvalues) is n. Berkolaiko [24] have shown that the number of nodal domains is bounded to the interval [n − l, n] where l is the minimal number of bonds which should be cut so that the resulting graph is a tree.…”

Section: A Short Introduction To Quantum Graphsmentioning

confidence: 99%

Nodal domains on isospectral quantum graphs: the resolution of isospectrality?

Band¹,

Shapira²,

Smilansky³

2006

J. Phys. A: Math. Gen.

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Abstract. We present and discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in R 2 which were introduced recently in [1-5] all based on Sunada's construction of isospectral domains[6]. After presenting some of the properties of these graphs, we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.

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“…In one dimension Sturm's oscillation theorem [2] states ν N = N under very general conditions. The generalisation of this seminal result to quasi one dimensional systems such as quantum graphs has been a recent research topic [3,4]. For arbitrary dimension a seminal result is Courant's nodal domain theorem [5] which states ν N ≤ N for the Laplacian in any dimension.…”

Section: Introductionmentioning

confidence: 99%

On the nodal count statistics for separable systems in any dimension

Gnutzmann¹,

Lois²

2013

J. Phys. A: Math. Theor.

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Abstract. We consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in arbitrary dimension. We give an explicit expression for the limiting distribution of normalised nodal counts and analyse some of its universal properties. Our results are illustrated by detailed discussion of simple examples and numerical nodal count distributions.

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Eigenvalue and nodal properties on quantum graph trees*

Cited by 28 publications

References 6 publications

The nodal count {0,1,2,3,…} implies the graph is a tree

The nodal count {0,1,2,3,…} implies the graph is a tree

Nodal domains on isospectral quantum graphs: the resolution of isospectrality?

On the nodal count statistics for separable systems in any dimension

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