Trace formulae for quantum graphs

Bölte, Jens; Endres, Sebastian

doi:10.1090/pspum/077/2459873

Cited by 12 publications

(13 citation statements)

References 24 publications

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“…We have already mentioned that the scattering approach described above is not the only way to define quantum graphs. The other frequently used definition is based on self-adjoint extensions of H in (8) defined on the Dirichlet domain H 0 (see [15] and references therein). A complete description of all possible self-adjoint extensions was given in [33].…”

Section: Definitionsmentioning

confidence: 99%

“…A complete description of all possible self-adjoint extensions was given in [33]. In general each self-adjoint extension is equivalent to energy-dependent matrices σ KS,i (k) relating the outgoing amplitudes to the incoming amplitudes of Ψ ∈ A(k) at each vertex i instead of (15). These matrices can then be grouped together to form a global unitary scattering matrix S KS (k) as in (18), and a global quantum map…”

Section: Definitionsmentioning

confidence: 99%

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Eigenfunction statistics on quantum graphs

2010

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We investigate the spatial statistics of the energy eigenfunctions on large quantum graphs. It has previously been conjectured that these should be described by a Gaussian Random Wave Model, by analogy with quantum chaotic systems, for which such a model was proposed by Berry in 1977. The autocorrelation functions we calculate for an individual quantum graph exhibit a universal component, which completely determines a Gaussian Random Wave Model, and a system-dependent deviation. This deviation depends on the graph only through its underlying classical dynamics. Classical criteria for quantum universality to be met asymptotically in the large graph limit (i.e. for the non-universal deviation to vanish) are then extracted. We use an exact field theoretic expression in terms of a variant of a supersymmetric σ model. A saddle-point analysis of this expression leads to the estimates. In particular, intensity correlations are used to discuss the possible equidistribution of the energy eigenfunctions in the large graph limit. When equidistribution is asymptotically realized, our theory predicts a rate of convergence that is a significant refinement of previous estimates. The universal and system-dependent components of intensity correlation functions are recovered by means of an exact trace formula which we analyse in the diagonal approximation, drawing in this way a parallel between the field theory and semiclassics. Our results provide the first instance where an asymptotic Gaussian Random Wave Model has been established microscopically for eigenfunctions in a system with no disorder.

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Section: Definitionsmentioning

confidence: 99%

Section: Definitionsmentioning

confidence: 99%

Eigenfunction statistics on quantum graphs

2010

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“…The statement of the trace formula in [12] is a lot more general than the one we give here, this special case with Kirchhoff boundary conditions can also be found in the survey paper [3].…”

Section: A Trace Formulamentioning

confidence: 63%

“…The relation between its spectrum or more generally the spectrum of a Schrödinger operator and the underlying graph is an active area of research. Various exact trace formulae relate the two, see for example [16], [12] or [3].…”

Section: ∂ ∂Xmentioning

confidence: 99%

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Recovering quantum graphs from their Bloch spectrum

Rueckriemen¹

2013

Ann. inst. Fourier

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We define the Bloch spectrum of a quantum graph to be the collection of the spectra of a family of Schrödinger operators parametrized by the cohomology of the quantum graph. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum completely determines planar 3-connected quantum graphs.I would like to thank my supervisor Carolyn Gordon for her extensive support. Without our frequent meetings and her numerous helpful and necessary suggestions this paper could not have been written.

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The Trace Formula

Kurasov

2023

Operator Theory: Advances and Applications

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Trace formulae for quantum graphs

Cited by 12 publications

References 24 publications

Eigenfunction statistics on quantum graphs

Eigenfunction statistics on quantum graphs

Recovering quantum graphs from their Bloch spectrum

The Trace Formula

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