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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 ..."
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Cited by 7 (4 self)
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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.
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 ..."
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Cited by 5 (2 self)
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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
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"... Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide ..."
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Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
In memoriam Vladimir A. Geyler (1943–2007) Numerical Simulation of Electron Scattering by Nanotube Junctions
, 2008
"... Abstract. We demonstrate the possibility of computing the intensity of electronic transport through various junctions of threedimensional metallic nanotubes. In particular, we observe that the magnetic field can be used to control the switch of electron in Ytype junctions. Keeping in mind the asym ..."
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Abstract. We demonstrate the possibility of computing the intensity of electronic transport through various junctions of threedimensional metallic nanotubes. In particular, we observe that the magnetic field can be used to control the switch of electron in Ytype junctions. Keeping in mind the asymptotic modeling of reliable nanostructures by quantum graphs, we conjecture that the scattering matrix of the graph should be the same as the scattering matrix of its nanosizeprototype. The numerical computation of the latter gives a method for determining the “gluing ” conditions at a graph. Exploring this conjecture, we show that the Kirchhoff conditions (which are commonly used on graphs) cannot be applied to model reliable junctions. This work is a natural extension of the paper [1], but it is written in a selfconsistent manner. DOI: 10.1134/S1061920808010020 1.