Quantum graphs: I. Some basic structures

By Peter Kuchment. 2004

649 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3bde0cdb886c7c71729ee1326b8dbcdbde6d0e02

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Citations in the corpus

Brownian motions on metric graphs (2011, background): “Metric graphs arise naturally as models in many domains, such as physics, chemistry, computer science and engineering to mention just a few — we refer the interested reader to [59] for a review of such models and for further references.”
Bound states of the NLS equation on metric graphs with localized nonlinearities (2015, background): “e graph G) is developed in [2, 3]. 4 2 Functionalsetting We start by recalling brieﬂy some basic facts and deﬁntions concerning metric graphs (for a modern account on the topic we refer the reader to [2, 17] and references therein). In this paper, by connected metric graph G = (V,E), we mean a conne”
Inverse spectral problem for quantum graphs (2006, background): “One can find a recent reference list with historical remarks in the book [2] and volumes [25, 26] devoted entirely to quantum graphs.”
A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric Graphs (2017, background): “A graph is a metric graph if each edge is assigned a positive length and if the graph is equipped with a natural metric where the distance between any two points of the graph (not necessarily vertices) is deﬁned to be the minimum length of all paths from one to the other [13].”
On the conservativeness and the recurrence of symmetric jump-diffusions (2012, background): “ND RECURRENCE OF JUMP-DIFFUSIONS 19 where E(c)[u] = Z E ∂u ∂x 2 dm, and E(j)[u] = X x,y∈V (u(x)−u(y))2 ω(x,y). The generators associated with E(c) and E(j) are called the quantum graph, see, e.g. [25] and the physical Laplacian, respectively. Let Fbe the closure of CLip 0 (X) with respect to the √”
An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter (2015, positive): “in the the proof of Lemma 2.1. Finally, substituting this estimate into the equality tβ(ψ,ψ) = Eβ we obtain (2.5). Finally, we will need a suitable form of the Sobolev inequality, see e.g. Lemma 8 in [24]: Lemma 2.3. For any 0 <ℓ≤ aand f∈ H1(0,a) there holds, with ξ∈ {0,a}, f(ξ)2 ≤ ℓ Za 0 f′(t)2d”
Existence of infinite stationary solutions of the $L^2$-subcritical and critical NLSE on compact metric graphs (2017, background): “here a full topological characterization of the existence of ground states. Let us brieﬂy recall some standard deﬁnitions and facts on compact metric graphs (for more details we refer for instance to [5, 14, 19]). Throughout the paper, a connected metric graph G = (V,E) denotes a connected metric sp”
SPECTRA OF SELF-ADJOINT EXTENSIONS AND APPLICATIONS TO SOLVABLE SCHRÖDINGER OPERATORS (2006, background): “Very important applications of the operator extension theory have been found recently in the physics of mesoscopic systems like heterostructures [72], quantum graphs [90, 91, 93 , 106] and circuits [1], quantum wells, dots, and wires [81].”
Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs (2004, methodology): “uits of quantum wires), waveguide theory, nanotechnology, dynamical systems, and photonic crystals. We will not discuss any details of these origins of quantum graphs, referring the reader instead to [10, 12, 19, 20, 23, 25, 26, 27, 28, 29, 32, 33, 34, 37] for further information, recent surveys, an”