Singular continuous and dense point spectrum
                    for sparse trees with finite dimensions

Breuer, Jonathan

doi:10.1090/crmp/042/03

Cited by 13 publications

(16 citation statements)

References 22 publications

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“…We argue in the spirit of [3,4,20] and suggest a quite general canonical form for the adjacency matrix of such graphs and supplement to the list of the examples. As a matter of fact, the algorithm applies not only to adjacency matrices, but to both Laplacians on graphs of such type.…”

Section: L Golinskiimentioning

confidence: 97%

“…The spectral theory of infinite graphs with one or several rays attached to certain finite graphs was initiated in [12,13,14,18] wherein a number of particular examples of unweighted (background) graphs is examined. We argue in the spirit of [3,4,20] and suggest a quite general canonical form for the adjacency matrix of such graphs and supplement to the list of the examples. As a matter of fact, the algorithm applies not only to adjacency matrices, but to both Laplacians on graphs of such type.…”

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confidence: 97%

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Spectra of infinite graphs with tails

Golinskiĭ

2016

Linear and Multilinear Algebra

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Abstract. We compute explicitly (modulo solutions of certain algebraic equations) the spectra of infinite graphs obtained by attaching one or several infinite paths to some vertices of certain finite graphs. The main result concerns a canonical form of the adjacency matrix of such infinite graphs. A complete answer is given in the case when the number of attached paths to each vertex is the same.Introduction and preliminaries 0.1. Graph theory. We begin with rudiments of the graph theory. For the sake of simplicity we restrict ourselves with simple, connected, undirected, finite or infinite (countable) weighted graphs, although the main result holds for weighted multigraphs and graphs with loops as well. We will label the vertex set V(Γ) by positive integers N = {1, 2, . . .}, {v} v∈V = {j} ω j=1 , ω ≤ ∞. The symbol i ∼ j means that the vertices i and j are incident, i.e., {i, j} belongs to the edge set E(Γ). A graph Γ is weighted if a positive number d ij (weight) is assigned to each edge {i, j} ∈ E(Γ). In case d ij = 1 for all i, j, the graph is unweighted.The degree (valency) of a vertex v ∈ V(Γ) is a number γ(v) of edges emanating from v. A graph Γ is said to be locally finite, if γ(v) < ∞ for all v ∈ V(Γ), and uniformly locally finite, if sup V γ(v) < ∞.The spectral graph theory deals with the study of spectra and spectral properties of certain matrices related to graphs (more precisely, operators generated by such matrices in the standard basis {e k } k∈N and acting in the corresponding Hilbert spaces C n or ℓ 2 = ℓ 2 (N)). One of the most notable of them is the adjacency matrix A(Γ)The corresponding adjacency operator will be denoted by the same symbol. It acts asClearly, A(Γ) is a symmetric, densely-defined linear operator, whose domain is the set of all finite linear combinations of the basis vectors. The operator Key words and phrases. Infinite graphs; adjacency operator; spectrum; Jacobi matrices of finite rank; Jost function. 2 L. GOLINSKIIA(Γ) is bounded and self-adjoint in ℓ 2 , as long as the graph Γ is uniformly locally finite.Whereas the spectral theory of finite graphs is very well established (see, e.g., [1,5,6,7]), the corresponding theory for infinite graphs is in its infancy. We refer to [16,17,21] for the basics of this theory. In contrast to the general consideration in [17], our goal is to carry out a complete spectral analysis (canonical models for the adjacency operators and computation of the spectrum) for a class of infinite graphs which loosely speaking can be called "finite graphs with tails attached to them". To make the notion precise, we define first an operation of coupling well known for finite graphs (see, e.g., [7, Theorem 2.12]).Definition 0.1. Let Γ k , k = 1, 2, be two weighted graphs with no common vertices, with the vertex sets and edge sets V(Γ k ) and E(Γ k ), respectively, and let v k ∈ V(Γ k ). A weighted graph Γ = Γ 1 + Γ 2 will be called a coupling by means of the bridgeSo we join Γ 2 to Γ 1 by the new edge of weight d between v 2 and v 1 .If the graph Γ 1 is fini...

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Section: L Golinskiimentioning

confidence: 97%

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confidence: 97%

Spectra of infinite graphs with tails

Golinskiĭ

2016

Linear and Multilinear Algebra

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“…We refer to [2][3][4][7][8][9][10][11]13,14,[20][21][22]29] for some related works on the spectral theory of operators on groups and graphs. Some of these articles put into evidence (Hecke-type) operators with large singular or singular continuous components.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis for convolution operators on locally compact groups

Măntoiu¹,

Aldecoa²

2007

Journal of Functional Analysis

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“…The techniques developed in this paper can be applied to improve some results due to Breuer [1,2], regarding the dimensionality of the spectral measure of the discrete Laplacian on spherically symmetric 6 sparse trees. 7 The spectral measure is purely singular and infinitely degenerated.…”

Section: )mentioning

confidence: 99%

“…(the essential spectrum contains the interval (−2, 2); see Theorem 3.8 of [2]), j = N l − N l−1 , and the local spectral dimension α of ρ, which is simply α(λ) ∈ 1 − ln r ln β − ε, 1 − ln r ln β + ε , 6 In the sense that any vertex of a given generation is connected to a fixed number of vertices from the next generation. 7 Sparse trees are trees which have arbitrarily long one-dimensional segments separated by occasional branchings.…”

Section: )mentioning

confidence: 99%