On the Weyl Law for Quantum Graphs

Odžak, Almasa; Šćeta, Lamija

doi:10.1007/s40840-017-0469-9

Cited by 7 publications

(6 citation statements)

References 19 publications

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“…For B in = 0 a proof can be found in [6,Prop. 4.2] (see also [28]). For B in = 0 bounded, claim (B.1) can be deduced by a perturbative argument.…”

Section: Appendix a Proof Of The Kreȋn Resolvent Formulaementioning

confidence: 99%

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

Cacciapuoti

2019

Symmetry

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We consider a compact metric graph of size ε, and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G ε we define an Hamiltonian H ε , properly scaled with the parameter ε. We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H ε (in a suitable norm resolvent sense) as ε → 0. The effective Hamiltonian depends on the spectral properties of an auxiliary ε-independent Hamiltonian defined on the compact graph obtained by setting ε = 1. If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit ε → 0, the leads are decoupled.

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“…For B in = 0 a proof can be found in [6,Prop. 4.2] (see also [28]). For B in = 0 bounded, claim (B.1) can be deduced by a perturbative argument.…”

Section: Appendix a Proof Of The Kreȋn Resolvent Formulaementioning

confidence: 99%

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

Cacciapuoti

2019

Symmetry

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“…σ(L) be the spectrum of L. We define λ + min = inf{λ : λ > 0 and λ ∈ σ(L)}, with the convention that inf ∅ = +∞. Following [24], we define…”

Section: Aq Andmentioning

confidence: 99%

“…Proof. The goal is to obtain a Weyl's law for L using Weyl's law from the Laplacian given in [24,Theorem 2]. To this end, we let…”

Section: Aq Andmentioning

confidence: 99%

Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs

Bolin¹,

Kovács²,

Kumar³

et al. 2023

Preprint

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The fractional differential equation L β u = f posed on a compact metric graph is considered, where β > 1 /4 and L = κ − d dx (H d dx ) is a secondorder elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients κ, H. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L −β . For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L 2 (Γ×Γ)error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for the example L = κ 2 − ∆, κ > 0 are performed to illustrate the theoretical results.

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“…But, secondly, as a notable by-product of our estimates, which are both sharp for each k and asymptotically sharp for each graph as k → ∞, we can obtain asymptotic relations of Weyl type for the spectral minimal energies which strongly recall the eigenvalue Weyl asymptotics. More precisely, we will show that the energies L k,p grow as (1.1) π 2 L 2 k 2 + O(k) as k → ∞, exactly like the eigenvalues of various realisations of the Laplacian on compact metric graphs [Nic87,CW05,BE09,OS19]. This may also be compared with the case of planar domains, where a two-dimensional analogue of (1.1) is conjectured for the (Dirichlet) spectral minimal energiesthe so-called hexagonal conjecture -but to date only two-sided asymptotic bounds are available (see [BNH17, § 10.9.1] and Remark 3.4).…”

Section: Introductionmentioning

confidence: 99%

Asymptotics and estimates for spectral minimal partitions of metric graphs

Hofmann¹,

Kennedy²,

Mugnolo³

et al. 2020

Preprint

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We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al (2020), arXiv:2005]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one for eigenvalues of quantum graph Laplacians with various vertex conditions. Drawing on two examples we show that in general no second term in the asymptotic expansion for minimal partition energies can exist, but also that various kinds of behaviour are possible. We also study certain aspects of the asymptotic behaviour of the minimal partitions themselves.

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On the Weyl Law for Quantum Graphs

Cited by 7 publications

References 19 publications

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs

Asymptotics and estimates for spectral minimal partitions of metric graphs

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