Results 1  10
of
26
Scattering solutions in a network of thin fibers: small diameter asymptotics, Preprint (mathph/0609021
, 2006
"... 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 ∈ Γ for the problems on the limiting graph. ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
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 ∈ Γ 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.
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 ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
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
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 eigenvalue ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
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.
ON THE SPECTRA OF CARBON NanoStructures
, 2007
"... An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nanostructures (including graphene and all types of singlewall nanotubes) is provided. ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nanostructures (including graphene and all types of singlewall nanotubes) is provided.
Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
 J. PHYS. A: MATH. THEOR. A
, 2007
"... In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achiev ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of δtype boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.
Laplace operator in networks of thin fibers: spectrum near the threshold
 IN STOCHASTIC ANALYSIS IN MATHEMATICAL PHYSICS, 69–93, WORLD SCI. PUBL
, 2008
"... Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. These results were published in [21]. The present paper contains a detailed review of [21] under some assumptions which make the ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. These results were published in [21]. The present paper contains a detailed review of [21] under some assumptions which make the results 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.
Approximations of singular vertex couplings in quantum graphs
, 2007
"... We discuss approximations of the vertex coupling on a starshaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edgepermutation symmetry is present. It is shown that the CheonShigehara technique using δ interactions with nonlinearly s ..."
Abstract

Cited by 6 (6 self)
 Add to MetaCart
We discuss approximations of the vertex coupling on a starshaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edgepermutation symmetry is present. It is shown that the CheonShigehara technique using δ interactions with nonlinearly scaled couplings yields a 2nparameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges one can approximate the `n+1 ´parameter family of all 2 timereversal invariant couplings.
Leaky quantum graphs: A review
, 2007
"... The aim of this review is to provide an overview of a recent work concerning “leaky ” quantum graphs described by Hamiltonians given formally by the expression − ∆ − αδ(x − Γ) with a singular attractive interaction supported by a graphlike set in R ν, ν = 2, 3. We will explain how such singular ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
The aim of this review is to provide an overview of a recent work concerning “leaky ” quantum graphs described by Hamiltonians given formally by the expression − ∆ − αδ(x − Γ) with a singular attractive interaction supported by a graphlike set in R ν, ν = 2, 3. We will explain how such singular Schrödinger operators can be properly defined for different codimensions of Γ. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of Γ influences their spectra and the scattering, strongcoupling asymptotic behavior, and a discrete counterpart to leakygraph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the
Convergence of resonances on thin branched quantum wave guides
 J. MATH. PHYS
, 2007
"... We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family Xε of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that th ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family Xε of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on Xε approximate those of the Laplacian with “free” boundary conditions on X0, the skeleton graph of Xε.