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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.
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
Solving the Helmholtz equation for membranes of arbitrary shape, sent to Journal of Physics A
, 2008
"... Abstract. I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of thi ..."
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Cited by 2 (1 self)
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Abstract. I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of this approach, I have considered a large number of examples, part of them taken from the literature, and part of them new. When possible, I have tested the accuracy of these results by comparing them with the exact results (when available) or with results from the literature. In particular, in the case of the Lshaped membrane, the first example discussed in the paper, I show that it is possible to extrapolate the results obtained with different grid sizes to obtain higly precise results. Finally, I also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results.
Contents
, 710
"... Abstract. 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, ..."
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Abstract. 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 various boundary conditions. We obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions, uniformly for k ≤ Cε −1, in terms of scattering data on a noncompact 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 (Gε). Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and use of a priori estimates to show that all
Scientific jury:
, 2012
"... The primary goal of the thesis is to study localization of Laplacian eigenfunctions in bounded domains when an eigenfunction is mainly supported by a small region of the domain and vanishing outside this region. The highfrequency and lowfrequency localization in simple and irregular domains has be ..."
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The primary goal of the thesis is to study localization of Laplacian eigenfunctions in bounded domains when an eigenfunction is mainly supported by a small region of the domain and vanishing outside this region. The highfrequency and lowfrequency localization in simple and irregular domains has been investigated for both Dirichlet and Neumann boundary conditions. Three types of highfrequency localization (whispering gallery, bouncing ball, and focusing eigemodes) have been revisited in circular, spherical and elliptical domains by deriving explicit inequalities on the norm of eigenfunctions. In turn, no localization has been found in most rectangular domains that led to formulating an open problem of characterization of domains that admit highfrequency localization. Using the Maslovtype differential inequalities, the exponential decay of lowfrequency Dirichlet eigenfunctions has been extensively studied in various domains with branches of variable crosssectional profiles. Under an explicit condition, the L2norm of an eigenfunction has been shown to exponentially decay along the branch with an explicitly computed decay rate. This rigorous upper bound, which is applicable in any dimension and for both finite and infinite branches, presents a new achievement in the theory of classical and quantum waveguides, with potential applications in microelectronics, optics and acoustics. For bounded quantum waveguides with constant crosssectional profiles, a sufficient condition on the
and
, 2008
"... In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. ..."
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In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding onshell Smatrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E> 0 is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff’s law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the onshell Smatrix of a graph is factorizable in terms of the Smatrices of its subgraphs. All proofs only use known facts from the theory of selfadjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.
and
, 2008
"... Dedicated to S.P. Novikov on the occasion of his 60th birthday In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the ..."
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Dedicated to S.P. Novikov on the occasion of his 60th birthday In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding onshell Smatrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E> 0 is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff’s law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the onshell Smatrix of a graph is factorizable in terms of the Smatrices of its subgraphs. All proofs only use known facts from the theory of selfadjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.
On the Spectrum of Curved Quantum Waveguides
, 2003
"... The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is ..."
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The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound on the gap between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved