Journal ArticleDOI
On the spectrum of the Laplacian on regular metric trees
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In this paper, the authors consider a special class of trees, namely the so-called regular metric trees and show that the space L 2 decomposes into the orthogonal sum of subspaces reducing the Laplacian operator Δ.Abstract:
A metric tree r is a tree whose edges are viewed as non-degenerate line segments. The Laplacian Δ on such a tree is the operator of second order differentiation on each edge, complemented by the Kirchhoff matching conditions at the vertices. The spectrum of Δ can be quite varied, reflecting the geometry of a tree. We consider a special class of trees, namely the so-called regular metric trees. Any such tree r possesses a rich group of symmetries. As a result, the space L 2 (Γ) decomposes into the orthogonal sum of subspaces reducing the operator Δ. This leads to detailed spectral analysis of Δ. We survey recent results on this subject.read more
Citations
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Journal ArticleDOI
Quantum graphs: I. Some basic structures
TL;DR: A quantum graph as discussed by the authors is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian, which is a special case of a combinatorial graph model.
Journal ArticleDOI
Quantum graphs: Applications to quantum chaos and universal spectral statistics
Sven Gnutzmann,Uzy Smilansky +1 more
TL;DR: In this paper, the spectral theory of quantum graphs is discussed and exact trace formulae for the spectrum and the quantum-to-classical correspondence are discussed, as well as its application to quantum chaos.
Journal ArticleDOI
Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
TL;DR: In this article, a Schnol-type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available.
Book
Spectral Analysis on Graph-like Spaces
TL;DR: In this article, the convergence results for star graphs and associated Laplacians were obtained for different Hilbert spaces and boundary triples, and two operators in different Hilbert space operators were compared.
Journal ArticleDOI
Inverse spectral problem for quantum graphs
Pavel Kurasov,Marlena Nowaczyk +1 more
TL;DR: In this paper, the inverse spectral problem for the Laplace operator on a finite metric graph is investigated and it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2.
References
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Journal ArticleDOI
Graph models for waves in thin structures
TL;DR: A brief survey on graph models for wave propagation in thin structures is presented in this article, with references to works with other studies are provided, although references are limited to spectral problems.
Book ChapterDOI
Probability on Trees: An Introductory Climb
TL;DR: In this paper, the authors define the first moment method and the second moment method for percolation on a connected graph, as well as the first-moment method of random walks.
Journal ArticleDOI
Eigenvalue Estimates for the Weighted Laplacian on Metric Trees
K. Naimark,Michael Solomyak +1 more
TL;DR: In this article, the eigenvalue of the Laplacian on a metric tree is studied and the spectral analysis of the problem is carried out for a particular class of trees and weights.
Journal ArticleDOI
Schrödinger operators on homogeneous metric trees: spectrum in gaps
TL;DR: In this paper, the spectral properties of the Schrodinger operator AgV = A0 + gV on a homogeneous rooted metric tree, with a decaying real-valued potential V and a coupling constant g ≥ 0, were studied.
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