A note on the Gaffney Laplacian on infinite metric graphs

Kostenko, Aleksey; Nicolussi, Noema

doi:10.1016/j.jfa.2021.109216

Cited by 4 publications

(10 citation statements)

References 18 publications

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“…More specifically, using the notion of finite volume graph ends introduced in [146], we are interested in conditions on the edge weights and under which finite volume graph ends serve as the proper boundary for Markovian extensions. Let us also mention that these results can be seen as the study of self-adjointness for the Gaffney Laplacian [148].…”

Section: Overview Of the Resultsmentioning

confidence: 92%

“…However, the operator defined on this domain has a different name -the Gaffney Laplacian -and it is not symmetric in general. Moreover, this operator is not always closed (see [148]).…”

Section: Gaffney Laplacianmentioning

confidence: 99%

“…Intuitively, this problem (as well as the self-adjoint uniqueness) is closely related to finding appropriate boundary notions for infinite graphs. For unweighted metric graphs, that is, with D Á 1, the question was studied in [146,148] .dx/ < 1 for some n. Otherwise has infinite volume. We denote the set of all finite volume ends by C 0 .G I / and equip it with the induced topology from the end space C.G /.…”

Section: Markovian Uniqueness and Graph Endsmentioning

confidence: 99%

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Laplacians on Infinite Graphs

Kostenko

Nicolussi

2023

Memoirs of the European Mathematical Society

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The main focus in this memoir is on Laplacians on both weighted graphs and weighted metric graphs. Let us emphasize that we consider infinite locally finite graphs and do not make any further geometric assumptions. Whereas the existing literature usually treats these two types of Laplacian operators separately, we approach them in a uniform manner in the present work and put particular emphasis on the relationship between them. One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Our central goal is twofold. First of all, we explore the relationships between these two objects by comparing their basic spectral (self-adjointness, spectral gap, etc.), parabolic (Markovian uniqueness, recurrence, stochastic completeness, etc.), and metric (quasi-isometries, intrinsic metrics, etc.) properties. In turn, we exploit these connections either to prove new results for Laplacians on metric graphs or to provide new proofs and perspective on the recent progress in weighted graph Laplacians. We also demonstrate our findings by considering several important classes of graphs (Cayley graphs, tessellations, and antitrees).

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Section: Overview Of the Resultsmentioning

confidence: 92%

“…However, the operator defined on this domain has a different name -the Gaffney Laplacian -and it is not symmetric in general. Moreover, this operator is not always closed (see [148]).…”

Section: Gaffney Laplacianmentioning

confidence: 99%

Section: Markovian Uniqueness and Graph Endsmentioning

confidence: 99%

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Laplacians on Infinite Graphs

Kostenko

Nicolussi

2023

Memoirs of the European Mathematical Society

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“…Furthermore, descriptions of certain non-self-adjoint operators on metric graphs appear in the literature [26]. Besides these continuum models on metric graphs, difference operators acting on the vertices of discrete graphs have a long history, and there is an intimate relation between operators on metric graphs and operators on discrete graphs, see [6,16,18,37,38], as well as [31] for an extensive overview on operators on discrete graphs.…”

Section: Introductionmentioning

confidence: 99%

Spectral Monotonicity for Schrödinger Operators on Metric Graphs

Rohleder

Seifert

2020

Discrete and Continuous Models in the Theory of Networks

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Spectral properties of Schrödinger operators on compact metric graphs are studied and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for δ and δ ′ couplings and demonstrate the spectral properties on many examples. Amongst other things, properties of the ground state eigenvalue and eigenfunction and the spectral behavior under various perturbations of the metric graph or the vertex conditions are considered.

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“…However, the operator defined on this domain has a different name -the Gaffney Laplacian -and it is not symmetric in general. Moreover, this operator is not always closed (see [48]).…”

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confidence: 99%

Laplacians on infinite graphs: discrete vs continuous

Kostenko¹,

Nicolussi²

2021

Preprint

Self Cite

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There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to several diverse branches of mathematics and mathematical physics. The existing literature usually treats these two Laplacian operators separately. In this overview, we will focus on the relationship between them (spectral and parabolic properties). Our main conceptual message is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides.

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A note on the Gaffney Laplacian on infinite metric graphs

Cited by 4 publications

References 18 publications

Laplacians on Infinite Graphs

Laplacians on Infinite Graphs

Spectral Monotonicity for Schrödinger Operators on Metric Graphs

Laplacians on infinite graphs: discrete vs continuous

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