Infinite quantum graphs

Kostenko, Aleksey; Malamud, M. M.; Neidhardt, Hagen; Exner, Pavel

doi:10.1134/s1064562417010136

Cited by 4 publications

(4 citation statements)

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“…These weights have extensively discussed in [25,37,51], in the context of spectral theory of quantum graphs. We refer to G = G m,μ constructed above as the weighted combinatorial graph underlying the metric graph G.…”

Section: Proof Since the Torsion Function V Solves The Equationmentioning

confidence: 99%

On torsional rigidity and ground-state energy of compact quantum graphs

Mugnolo

Plümer

2022

Calc. Var.

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We develop the theory of torsional rigidity—a quantity routinely considered for Dirichlet Laplacians on bounded planar domains—for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to Pólya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in terms of its torsional rigidity: this is particularly attractive since computing the torsional rigidity reduces to inverting a matrix whose size is the number of the graph’s vertices and is, thus, much easier than computing eigenvalues.

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Section: Proof Since the Torsion Function V Solves The Equationmentioning

confidence: 99%

On torsional rigidity and ground-state energy of compact quantum graphs

Mugnolo

Plümer

2022

Calc. Var.

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“…Theorem 2.11 and also [6,15,11,58]). Moreover, it was observed recently in [26,44] by using the ideas from [43] that spectral properties of the operator H are closely connected with the corresponding properties of the discrete Laplacian defined in ℓ 2 (V; m) by the expression…”

Section: Introductionmentioning

confidence: 95%

“…One of the standard conditions to ensure the essential selfadjointness of H 0 is the existence of a positive lower bound on the edges lengths, ℓ * (G) = inf e∈E |e| > 0 (see [10]). Only recently several self-adjointness conditions without this rather restrictive assumption have been established in [26], [44] (see Section 2.3 for further details). Of course, the next natural question is the structure of the spectrum of the operator H. Clearly, the spectrum of an infinite quantum graph is not necessarily discrete and hence one is interested in the location of the bottom of the spectrum, λ 0 (H), as well as of the bottom of the essential spectrum, λ ess 0 (H), of H. Since the graph is infinite, many quantities of interest for finite quantum graphs (e.g., the number of vertices, edges, or its total length) are no longer suitable for these purposes and the corresponding bounds usually lead to trivial estimates.…”

Section: Introductionmentioning

confidence: 99%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

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We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.2010 Mathematics Subject Classification. Primary 34B45; Secondary 35P15; 81Q35.

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“…This includes local perturbations [Al97,AF00], random settings [Kl98, AW11, AW13, FLSSS, FHH12, Sh15] (see also the references therein), and quantum ergodicity regimes [AL15]. For complementary results, we refer the reader to the papers [Br07, BK13, Ro06a, Ro06b, RR07], and for the relationships between the Laplace operator on trees and quantum graphs, see [KMNE17]. However, it seems that resonances have not been systematically studied in the context of (regular) trees.…”

Section: Introductionmentioning

confidence: 99%