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25
Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide
, 2007
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APPROXIMATION OF QUANTUM GRAPH VERTEX COUPLINGS BY SCALED SCHRÖDINGER OPERATORS ON THIN BRANCHED MANIFOLDS
"... Abstract. We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate nontrivial vertex couplings. The lat ..."
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Cited by 17 (8 self)
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Abstract. We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate nontrivial vertex couplings. The latter include not only the δcouplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric δ ′couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal. 1.
An approximation to δ′ couplings on graphs
, 2004
"... We discuss a general parametrization for vertices of quantum graphs and show, in particular, how the δ′ s and δ′ coupling at an n edge vertex can be approximated by means of n+1 couplings of the δ type provided the latter are properly scaled. ..."
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Cited by 13 (2 self)
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We discuss a general parametrization for vertices of quantum graphs and show, in particular, how the δ′ s and δ′ coupling at an n edge vertex can be approximated by means of n+1 couplings of the δ type provided the latter are properly scaled.
Solvable models for the Schrödinger operators with δ ′ like potentials
"... Abstract. We turn back to the well known problem of interpretation of the Schrödinger operator with the pseudopotential αδ ′ (x), where δ(x) is the Dirac function and α ∈ R. We show that the problem in its conventional formulation contains hidden parameters and the choice of the proper selfadjoint o ..."
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Cited by 11 (1 self)
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Abstract. We turn back to the well known problem of interpretation of the Schrödinger operator with the pseudopotential αδ ′ (x), where δ(x) is the Dirac function and α ∈ R. We show that the problem in its conventional formulation contains hidden parameters and the choice of the proper selfadjoint operator is ambiguously determined. We study the asymptotic behavior of spectra and eigenvectors of the Hamiltonians with increasing smooth potentials perturbed by shortrange potentials αε −2 Ψ(ε −1 x). Appropriate solvable models are constructed and the corresponding approximation theorems are proved. We introduce the concepts of the resonance set ΣΨ and the coupling function θΨ: ΣΨ → R, which are spectral characteristics of the potential Ψ. The selfadjoint operators in the solvable models are determined by means of the resonance set and the coupling function.
Large Gaps in PointCoupled Periodic Systems of Manifolds
, 2002
"... We study a free quantum motion on periodically structured manifolds composed of elementary twodimensional "cells" connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components ar ..."
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Cited by 11 (7 self)
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We study a free quantum motion on periodically structured manifolds composed of elementary twodimensional "cells" connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or twodimensional squarelattice "carpets". We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies. 1
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 ..."
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Cited by 7 (6 self)
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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.
Spectral asymptotics for Schrödinger operators with periodic point interactions
 J. Math. Anal. Appl
, 2002
"... Spectrum of the secondorder differential operator with periodic point interactions in L2R is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms ..."
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Cited by 6 (0 self)
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Spectrum of the secondorder differential operator with periodic point interactions in L2R is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spectrum of the operators with anomalous spectral asymptotics (when the ratio between the lengths of the bands and gaps tends to zero at infinity) can be approximated by standard periodic “weighted ” operators with stepwise density functions. It is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal (generalized) resolvent of the periodic “Schrödinger operator ” with certain energydependent boundary conditions. The operator acting in an extended Hilbert space such that its resolvent restricted to L2R coincides with the formal resolvent is constructed explicitly. 2002 Elsevier Science Key Words: point interactions; spectral asymptotics; selfadjoint extensions.
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 6 (3 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