Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

Band, Ram; Berkolaiko, Gregory; Smilansky, Uzy

doi:10.1007/s00023-011-0124-1

Cited by 21 publications

(39 citation statements)

References 42 publications

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“…The following Theorem is an amalgamation of several results appearing in [40,3] and also some new results. We consider the matrix Z as a function on the torus by replacing k l with κ.…”

Section: 1mentioning

confidence: 85%

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Nodal Statistics on Quantum Graphs

Alon

Band

Berkolaiko

2018

Commun. Math. Phys.

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It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the "nodal surplus") for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graph's first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form -it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.

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“…The following Theorem is an amalgamation of several results appearing in [40,3] and also some new results. We consider the matrix Z as a function on the torus by replacing k l with κ.…”

Section: 1mentioning

confidence: 85%

“…Scattering from a graph. One can probe spectral properties of a graph by attaching (infinite) leads to it and considering the scattering of plane waves coming in from infinity [29,44,40,45,24,23,3].…”

Section: 1mentioning

confidence: 99%

Nodal Statistics on Quantum Graphs

Alon

Band

Berkolaiko

2018

Commun. Math. Phys.

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“…A trace formula for the spectral counting function of a metric graph [50] was used to solve the inverse spectral problem on metric graphs [63]. It was suggested by Smilansky that similar trace formulae exist for functions of the nodal count and there is indeed some supporting evidence and derivations for specific classes of two-dimensional surfaces in [8,9,11], and progress on the problem for metric graphs in [10]. Yet, an exact trace formula for the nodal count has not been found yet.…”

Section: Discussionmentioning

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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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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“…The continuity condition is empty at every vertex since there is only one edge. The current conservation condition at the vertex 0 becomes (4) f (0) = 0, and at the vertex L becomes (5) − f (L) = 0.…”

Section: Schrödinger Equation On a Metric Graphmentioning

confidence: 99%

“…Exercise 2.1. Solve the eigenvalue equation with λ < 0 and show that the vertex conditions (4) and (5) are never satisfied simultaneously (ignore the trivial solution f (x) ≡ 0). Exercise 2.2.…”

Section: Schrödinger Equation On a Metric Graphmentioning

confidence: 99%