Relationship between scattering matrix and spectrum of quantum graphs

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-t asymptotics for the trace of the heat kernel.

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“…The following statement is a generalisation of a result found in [5,21] that is valid for k-independent S-matrices.…”

Section: The Secular Equationsupporting

confidence: 62%

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions

Bölte

Endres

2009

Ann. Henri Poincaré

106

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“…To get an analytic handle on the nodal distribution we combine two techniques of quantum graphs analysis: the magnetic-nodal connection of Berkolaiko, Colin de Verdière and Weyand [10,19,15] and the secular manifold of Barra and Gaspard [7] further developed in [16,2,20]. We lay out the required foundations in the next subsections.…”

Section: Defining the Distributionmentioning

confidence: 99%

“…The main idea of Barra and Gaspard [7] was that if one wants to calculate the average of a certain function of the spectrum of a quantum graph, it is often possible to redefine this function in terms of the κ torus coordinates instead and then integrate over the secular manifold Σ with an appropriate measure. This idea was applied to eigenvalue statistics in the original paper [7], used to study eigenfunction statistics [16], eigenfunction scarring [20], band-gap statistics of periodic structures [2,54,26] and statistics of topological resonances [21]. Definition 3.10 (Barra-Gaspard measure [7,20]).…”

Section: 5mentioning

confidence: 99%

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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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“…In that paper, instead of studying directly the high energy behaviour of the eigenfunctions

ψ_{n}

of the quantum graph, the authors study the eigenfunctions

ϕ_{j} false(λ_{n} false)

of associated unitary operators

U (λ_{n})

which encode the classical evolution. It is shown that both notions of quantum ergodicity (for

ψ_{n}

ϕ_{j} false(λ_{n} false)

) are intimately related if the size of the graph goes to infinity, see also for a precise statement.…”

Section: Introductionmentioning

confidence: 99%

Quantum ergodicity for large equilateral quantum graphs

Ingremeau

Sabri

Winn

2019

J. London Math. Soc.

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Consider a sequence of finite regular graphs converging, in the sense of Benjamini–Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non‐zero coupling constant α) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case α=0 and U=0, the limit measure is the uniform measure on the edges. In general, it has an explicit C1 density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations.

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Relationship between scattering matrix and spectrum of quantum graphs

Cited by 30 publications

References 41 publications

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions

Nodal Statistics on Quantum Graphs

Quantum ergodicity for large equilateral quantum graphs

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