Journal ArticleDOI
Quantum graphs: I. Some basic structures
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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.Abstract:
A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.read more
Citations
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Journal ArticleDOI
Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)
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.
Journal ArticleDOI
Spectra of self-adjoint extensions and applications to solvable schrödinger operators
TL;DR: In this article, a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples is given, and a description of the spectra of selfadjoint extension in terms of the corresponding Krein maps (Weyl functions) is given.
Book
Semigroup Methods for Evolution Equations on Networks
TL;DR: In this article, a crash course in Cortical Modeling is described, with a focus on the evolution of self-adjoint operators in the context of function spaces on networks.
References
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Book
Spectral Graph Theory
TL;DR: Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigen values and quasi-randomness
MonographDOI
Algebraic graph theory
TL;DR: In this article, the authors introduce algebraic graph theory and show that the spectrum of a graph can be modelled as a graph graph, and the spectrum can be represented as a set of connected spanning trees.
Book
Theory of linear operators in Hilbert space
N. I. Akhiezer,I. M. Glazman +1 more
TL;DR: In this article, the main properties of bounded and unbounded operators, adjoint operators, symmetric and self-adjoint operators in hilbert spaces are discussed, as well as the stability of self-jointness under small perturbations.
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