text: Let (G e ) e>0 be a family of ‘e-thin’ Riemannian manifolds modeled on a finite metric graph G, for example, the e-neighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on G e , as e→0, for various boundary conditions. We obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions, uniformly for k≤Ce −1 , in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family (G e ). Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and the use of a priori estimates to show that all eigenfunctions are obtained in this way.
  @cite_14: We consider a family of open sets M? which shrinks with respect to an appropriate parameter ? to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of M? 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.
  @cite_4: Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph Γ . We calculate the Lagrangian gluing conditions at vertices ( v ) for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network.
  @cite_8: 
  @cite_3: Let M be a planar embedded graph whose arcs meet transversally at the vertices. Let ?(ɛ) be a strip-shaped domain around M, of width ɛ except in a neighborhood of the singular points. Assume that the boundary of ?(ɛ) is smooth. We define comparison operators between functions on ?(ɛ) and on M, and we derive energy estimates for the compared functions. We define a Laplace operator on M which is in a certain sense the limit of the Laplace operator on ?(ɛ) with Neumann boundary conditions. In particular, we show that the p-th eigenvalue of the Laplacian on ?(ɛ) converges to the p-th eigenvalue of the Laplacian on M as ɛ tends to 0. A similar result holds for the magnetic Schrodinger operator.
  @cite_0: Abstract Let M be a finite graph in the plane and let Me be a domain that looks like the e-fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on Me converges when e → 0 to the spectrum of an ODE problem on M. The 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.
  @cite_19: 
  @cite_5: Our talk at Lisbon SAMP conference was based mainly on our recent results (published in Comm. Math. Phys.) on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. The present paper contains a detailed review of these results under some assumptions which make them much more transparent. It also contains several new theorems on the structure of the spectrum near the threshold. small diameter asymptotics of the resolvent, and solutions of the evolution equation.
  @cite_13: 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 nontrivial 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.
  @cite_17: 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 Laplace–Beltrami 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 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.
summary: As already mentioned, the Neumann problem was treated in @cite_19 , @cite_8 , @cite_0 , KucZen:ASNLTD @cite_3 , @cite_17 . For Dirichlet boundary conditions, Post @cite_14 derived the first two terms of in the case of 'small' vertex neighborhoods, see Theorem . In the recent preprint @cite_5 Molchanov and Vainberg study the Dirichlet problem and show that, in the context of Theorem , the @math converge to eigenvalues of the quantum graph described in Theorem ; this was conjectured in @cite_4 , where also some results on the scattering theory on non-compact graphs are obtained. However, their statements are unclear as to whether the multiplicities coincide; also, they do not consider the effect of @math eigenvalues on @math or uniform asymptotics for large @math . In @cite_13 a related model is considered. The method in the previously cited papers is to compare quadratic forms or to show resolvent convergence of some sort, and in all cases only the leading asymptotic behavior is obtained.