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42
Convergence of Spectra of Mesoscopic Systems Collapsing Onto a Graph.
 J. Math. Anal. Appl
, 1999
"... Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic f ..."
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Cited by 55 (2 self)
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Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic field is also allowed. Considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry. The results of the paper extend the ones previously obtained by J. Rubinstein and M. Schatzman. 2000 MSC: 35Q40, 35P15, 35J10, 81V99 Key words and phrases: mesoscopic system, Schrodinger operator, spectrum 1 Introduction In recent years one has witnessed growing interest in spectral theory of differential (versus difference) operators on graphs. Although probably one of the first such studies was done in physical chemistry [47], the main thrust 1 in this direction came from the mesoscopic physics [29]. Recent progress in nanotechnology and microelectronics en...
Convergence of spectra of graphlike thin manifolds
 J. Geom. Phys
"... Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at th ..."
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Cited by 37 (14 self)
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Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices. 1.
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
 Journal of Physics A: Mathematical and General
"... Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) ..."
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Cited by 27 (7 self)
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Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature. 1.
The Generalized Star Product And The Factorization Of Scattering Matrices On Graphs
 J. Math. Phys
, 2000
"... . In this article we continue our analysis of Schrodinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star p ..."
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Cited by 20 (7 self)
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. In this article we continue our analysis of Schrodinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discussed. 1. INTRODUCTION Potential scattering for one particle Schrodinger operators on the line possesses a remarkable property concerning its (onshell) scattering matrix given as a 2 2 matrixvalued function of the energy. Let the potential V be given as the sum of two potentials V 1 and V 2 with disjoint support. Then the scattering matrix for V at a given energy is obtained from the two scattering matrices for V 1 and V 2 at the same energy by a certain nonlinear, nonco...
Spectra of Graph Neighborhoods and Scattering
"... Let (Gε)ε>0 be a family of ’εthin’ Riemannian manifolds modeled on a finite metric graph G, for example, the εneighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the LaplaceBeltrami operator on Gε as ε → 0, for ..."
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Cited by 14 (3 self)
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Let (Gε)ε>0 be a family of ’εthin’ Riemannian manifolds modeled on a finite metric graph G, for example, the εneighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the LaplaceBeltrami operator on Gε as ε → 0, for
Spectral properties of Schrödinger operators with a strongly attractive δ interaction supported by a surface
 Proceedings of the NSF Summer Research Conference (Mt. Holyoke 2002); AMS “Contemporary Mathematics” Series
, 2003
"... Abstract. We investigate the operator − ∆ − αδ(x − Γ) in L 2 (R 3), where Γ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as α → ∞ which involves a “twodimensional ” compar ..."
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Cited by 13 (4 self)
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Abstract. We investigate the operator − ∆ − αδ(x − Γ) in L 2 (R 3), where Γ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as α → ∞ which involves a “twodimensional ” comparison operator determined by the geometry of the surface Γ. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic Γ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem. 1.
The inverse scattering problem for metric graphs and the traveling salesman problem
, 2006
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Coupling in the singular limit of thin quantum waveguides
 J. Math. Phys
"... Abstract. We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by ..."
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Cited by 12 (3 self)
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Abstract. We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e. a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a non trivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling. 1.
Anderson localization for radial treelike random quantum graphs
, 2008
"... We prove that certain random models associated with radial, treelike, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family ..."
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Cited by 10 (0 self)
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We prove that certain random models associated with radial, treelike, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.
On the spectrum of the Dirichlet Laplacian in a narrow strip
 Israeli Math. J
"... Abstract. This is a continuation of the paper [3]. We consider the Dirichlet Laplacian in a family of unbounded domains {x ∈ R, 0 < y < ǫh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We show that the number of eigenvalues lyin ..."
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Cited by 10 (0 self)
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Abstract. This is a continuation of the paper [3]. We consider the Dirichlet Laplacian in a family of unbounded domains {x ∈ R, 0 < y < ǫh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We show that the number of eigenvalues lying below the essential spectrum indefinitely grows as ǫ → 0, and find the twoterm asymptotics in ǫ → 0 of each eigenvalue and the oneterm asymptotics of the corresponding eigenfunction. The asymptotic formulae obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on R that depends only on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator ∆ǫ. 1.