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23
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.
Heat kernels on metric graphs and a trace formula
, 2007
"... We study heat semigroups generated by selfadjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kerne ..."
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Cited by 16 (2 self)
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We study heat semigroups generated by selfadjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.
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
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.
The inverse scattering problem for metric graphs and the traveling salesman problem
, 2006
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Spectral convergence of noncompact quasionedimensional spaces
 Ann. H. Poincaré
"... Abstract. We consider a family of noncompact manifolds Xε (“graphlike manifolds”) approaching a metric graph X0 and establish convergence results of the related natural operators, namely the (Neumann) Laplacian ∆ Xε and the generalised Neumann (Kirchhoff) Laplacian ∆ X0 on the metric graph. In par ..."
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Cited by 8 (0 self)
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Abstract. We consider a family of noncompact manifolds Xε (“graphlike manifolds”) approaching a metric graph X0 and establish convergence results of the related natural operators, namely the (Neumann) Laplacian ∆ Xε and the generalised Neumann (Kirchhoff) Laplacian ∆ X0 on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations. 1.
Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
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Cited by 8 (5 self)
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Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
Zigzag periodic nanotube in magnetic field
, 2006
"... We consider the magnetic Schrödinger operator on the socalled zigzag periodic metric graph (a quasi 1D continuous model of zigzag nanotubes) with a periodic potential. The magnetic field (with the amplitude B ∈ R) is uniform and it is parallel to the axis of the nanotube. The spectrum of this opera ..."
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Cited by 7 (5 self)
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We consider the magnetic Schrödinger operator on the socalled zigzag periodic metric graph (a quasi 1D continuous model of zigzag nanotubes) with a periodic potential. The magnetic field (with the amplitude B ∈ R) is uniform and it is parallel to the axis of the nanotube. The spectrum of this operator consists of an absolutely continuous part (spectral bands separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that endpoints of the gaps are periodic or antiperiodic eigenvalues or resonances (real branch points of the Lyapunov function). We describe the spectrum as functions of B. For example, if B → Bk,m = π πk
A lower bound for nodal count on discrete and metric graphs
, 2006
"... Abstract. We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we show that the number of nodal domains of the nth eigenfunction is bounded below by n − ℓ, ..."
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Cited by 7 (1 self)
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Abstract. We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we show that the number of nodal domains of the nth eigenfunction is bounded below by n − ℓ, where ℓ is the number of links that distinguish the graph from a tree. Our results apply to operators on both discrete (combinatorial) and metric (quantum) graphs. They complement already known analogues of a result by Courant who proved the upper bound n for the number of nodal domains. To illustrate that the genericity condition is essential we show that if it is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction. In the appendix we review the proof of the case ℓ = 0 on metric trees which has been obtained by other authors. 1.