Results 11  20
of
28
First order approach and index theorems for discrete and metric graphs
 Ann. Henri Poincaré
, 2009
"... Abstract. The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (“quantum”) graphs. In order to cover all selfadjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discr ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (“quantum”) graphs. In order to cover all selfadjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators acting on a decorated graph. The decoration at each vertex of degree d is given by a subspace of C d, generalising the fact that a function on the standard vertex space has only a scalar value. We develop the notion of exterior derivative, differential forms, Dirac and Laplace operators in the discrete and metric case, using a supersymmetric framework. We calculate the (supersymmetric) index of the discrete Dirac operator generalising the standard index formula involving the Euler characteristic of a graph. Finally, we show that the corresponding index for the metric Dirac operator agrees with the discrete one. 1.
Propagation of Waves in Networks of Thin Fibers
, 902
"... The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can desc ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph. Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it. 1
Graphlike models for thin waveguides with Robin boundary conditions
"... Abstract. We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions in view of the approximating problem for a Quantum Graph. We prove that the projections on each transverse mode generically give rise to decoupling conditions while exceptionally in ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions in view of the approximating problem for a Quantum Graph. We prove that the projections on each transverse mode generically give rise to decoupling conditions while exceptionally in the initial domain one can have non decoupling conditions in the vertex. The non decoupling conditions are related to the existence of zero energy states on the threshold of the continuum spectrum. 1.
A GENERAL APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS
"... Abstract. We demonstrate that any selfadjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schrödinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local chang ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We demonstrate that any selfadjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schrödinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently ‘lifted ’ to the manifold. For the corresponding operator a normresolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero. 1.
Proceedings of Symposia in Pure Mathematics
, 802
"... Thin tubes in mathematical physics, global analysis and spectral geometry ..."
Abstract
 Add to MetaCart
Thin tubes in mathematical physics, global analysis and spectral geometry
ON THE SPECTRUM OF THE DIRICHLET LAPLACIAN IN A NARROW STRIP,
, 705
"... There are several reasons why the study of the spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work [3], Colin de Verdière ..."
Abstract
 Add to MetaCart
There are several reasons why the study of the spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work [3], Colin de Verdière
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 → ..."
Abstract
 Add to MetaCart
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
Quantum networks modelled by graphs
, 706
"... Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of se ..."
Abstract
 Add to MetaCart
Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs ” — in other words, suitable families of shrinking manifolds — and discuss convergence of the spectra and resonances in such a setting.
unknown title
, 2006
"... Scattering solutions in a network of thin fibers: small diameter asymptotics. ..."
Abstract
 Add to MetaCart
Scattering solutions in a network of thin fibers: small diameter asymptotics.