Quantum graphs on radially symmetric antitrees

Kostenko, Aleksey; Nicolussi, Noema

doi:10.48550/arxiv.1901.05404

Cited by 5 publications

(13 citation statements)

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“…While antitrees first appeared in connection with random walks [18,43,59], they were actively studied from various different perspectives in the last years (see [7,15,45] for quantum graphs and [14, Section 2] for further references).…”

Section: Deficiency Indices Of Antitreesmentioning

confidence: 99%

Self-adjoint and Markovian extensions of infinite quantum graphs

Kostenko,

Mugnolo,

Nicolussi

2019

Preprint

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We investigate self-adjoint extensions of the minimal Kirchhoff Laplacian on an infinite metric graph. More specifically, the main focus is on the relationship between graph ends and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian H 0 . First, we introduce the notion of finite and infinite volume for (topological) ends of a metric graph and then establish a lower bound on the deficiency indices of H 0 in terms of the number of finite volume graph ends. This estimate is sharp and we also find a necessary and sufficient condition for the equality between the number of finite volume graph ends and the deficiency indices of H 0 to hold. Moreover, it turns out that finite volume graph ends play a crucial role in the study of Markovian extensions of H 0 . In particular, we show that the minimal Kirchhoff Laplacian admits a unique Markovian extension exactly when every topological end of the underlying metric graph has infinite volume. In the case of finitely many finite volume ends (for instance, the latter includes Cayley graphs of a large class of finitely generated infinite groups) we are even able to provide a complete description of all Markovian extensions of H 0 .

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Section: Deficiency Indices Of Antitreesmentioning

confidence: 99%

Self-adjoint and Markovian extensions of infinite quantum graphs

Kostenko,

Mugnolo,

Nicolussi

2019

Preprint

Self Cite

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“…The Kirchhoff Laplacian H is unitarily equivalent, cf. [KN,Theorem 3.5], to the following direct sum…”

Section: Resultsmentioning

confidence: 99%

“…For each n ∈ Z let S n denote a set of s n distinct vertices. Let Γ be the graph with the set of vertices V := ∪ n∈Z S n and such that two vertices u, v ∈ V are adjacent if and only if (u, v) ∈ The proof of this assertion is similar to that of [KN,Theorem 3.5]. Let us highlight the main ingredient of the proof by example.…”

Section: Spectral Decomposition Of the Two-sided Fibonacci Graphmentioning

confidence: 95%

“…Such a graph consists of tiles and line segments as shown on Figure 2. One has (5.8) where F sym consists of L 2 (Γ) functions that are horizontally symmetric on each tile (for precise definition see [KN,(2.11)] with "n ≥ 0" replaced by "n ∈ Z", and s n ∈ {1, 2}). H 2 n .…”

Section: Spectral Decomposition Of the Two-sided Fibonacci Graphmentioning

confidence: 99%

“…The main goal of this paper is to prove that the spectrum of H is a zero Lebesgue measure set whenever the sequence of sphere numbers satisfies certain dynamical conditions. Due to the large number of symmetries of Γ, the operator H can be written, see [KN,Theorem 3.5], as the direct sum of self-adjoint realizations of the following Sturm-Liouville differential expression…”

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Zero Measure and Singular Continuous Spectra for Quantum Graphs

Damanik,

Fang,

Sukhtaiev

2019

Preprint

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We introduce a dynamically defined class of unbounded, connected, equilateral metric graphs on which the Kirchhoff Laplacian has zero Lebesgue measure spectrum and a nontrivial singular continuous part. A new local Borg-Marchenko uniqueness result is obtained in order to utilize Kotani theory for aperiodic subshifts satisfying Boshernitzan's condition. Contents 1. Introduction 1 1.1. Overview 1 1.2. Setup 2 2. Main Results 4 3. Elements of Weyl-Titchmarsh Theory 6 3.1. Borg-Marchenko Theorem For Sturm-Liouville Operators in Impedance Form 6 3.2. Continuity of m-functions With Respect to Jumps. 11 3.3. Restriction Maps 12 4. Ergodic Sturm-Liouville Operators in Impedance Form 13 4.1. Dynamical Description of The Spectrum 14 4.2. Cantor Spectrum For the Full Line Model 17 5. Spectral Decomposition of the Two-sided Fibonacci Graph 18 References 20 D.D.

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On the Decomposition of the Laplacian on Metric Graphs

Breuer

Levi

2020

Ann. Henri Poincaré

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We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to the structure of the graph. Such decompositions have been extremely useful in the study of Schrödinger operators on metric trees. We show that the tree structure is not essential, and moreover, obtain a direct and simple correspondence between such decompositions in the discrete and continuum case.operators whose structure is directly related to the structure of the tree and to the boundary conditions at the vertices. A similar method exists in the discrete case, where the graph is considered as a combinatorial object and the operator studied is the discrete Laplacian or the adjacency matrix (see, e.g., [3,9,10,12]). While the similarity between the decomposition in the continuous and discrete case is clear and lies in the exploitation of the symmetry properties of the graph, it is important to note there are essential technical differences. Whereas the discrete case involves studying cyclic subspaces generated by specially chosen functions, the continuum case (as presented in [28,32]) involves defining the relevant invariant subspaces directly and relies heavily on the structure of the tree.It has recently been realized in [13] that in the discrete case, the tree structure is not essential for this decomposition. It is in fact possible to carry out this procedure for a more general class of graphs that we call 'family preserving' and whose definition we give below (see Definition 2.2) 1 . This class contains radially symmetric trees, antitrees (see Section 5 for the definition), and various other graphs.We refer the reader to [13] for examples and some graphics.The objective of this work is to extend this decomposition for family preserving graphs to the continuum case of metric graphs as well. Since, as remarked above, the standard decomposition technique for metric trees relies heavily on the tree structure, one needs to take a different approach. A natural approach to this task, and the one that we shall take, is to try and obtain a direct translation of the discrete decomposition to the decomposition in the metric case. Such an approach has also the added bonus of making explicit the connection between the combinatorial and continuum decompositions, and as we shall show, can recover the original Naimark-Solomyak procedure from the procedure in the discrete case.While employing the discrete decomposition scheme is indeed natural for getting a decomposition in the metric case, the results in this paper should by no means be viewed as a direct extension of those in [7]. First, as the functions in the metric case live on edges, whereas those in the discrete case live on vertices, it is a non-trivial task to adapt one procedure to the other. Second, as is evident in Section 4 (where we describe the decomposition algorithm) the discrete decomposition is only one of several steps...

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Quantum graphs on radially symmetric antitrees

Cited by 5 publications

References 29 publications

Self-adjoint and Markovian extensions of infinite quantum graphs

Self-adjoint and Markovian extensions of infinite quantum graphs

Zero Measure and Singular Continuous Spectra for Quantum Graphs

On the Decomposition of the Laplacian on Metric Graphs

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