Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

Breuer, Jonathan

doi:10.1007/s00220-006-0121-2

Cited by 32 publications

(51 citation statements)

References 15 publications

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“…This paper extends and complements the paper [5] in which the notion of sparse trees was introduced. Sparse trees are trees which have arbitrarily long 'one-dimensional' segments (by which we mean -intervals of Z), separated by occasional non-trivial branchings.…”

Section: Introductionsupporting

confidence: 60%

“…In addition to the new result described above, we also use this opportunity to expand the discussion on the basic setting and on some of the examples presented in [5]. Some basic facts that were briefly mentioned in that paper (such as the self-adjointness of the Laplacian on normal sparse trees), will be explained here in greater detail.…”

Section: Introductionmentioning

confidence: 94%

“…In this context, the family of sparse trees is interesting in that, when varying two parameter sequences (namely -the branching size and the distances between branchings), one encounters a rich spectrum of phenomena. We note, in particular, the existence of examples with spectral measures of fractional Hausdorff dimensions (see [5] and theorem 4.4 below). This paper is structured as follows.…”

Section: Introductionmentioning

confidence: 99%

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

Breuer¹

2007

Probability and Mathematical Physics

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Section: Introductionsupporting

confidence: 60%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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

Breuer¹

2007

Probability and Mathematical Physics

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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: 96%

“…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: 96%

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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