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33
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 ..."
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Cited by 8 (2 self)
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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.
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 ..."
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Cited by 7 (0 self)
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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.
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.
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 ..."
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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.
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
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 ..."
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Cited by 4 (1 self)
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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.
Vertex coupling in quantum graphs: approximations by scaled Schrödinger operators
, 2010
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On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon
 Nanosyst. Phys. Chem. Math
"... Let Ω ⊂ R2 be the exterior of a convex polygon whose side lengths are `1,..., `M. For a real constant α, let HΩα denote the Laplacian in Ω, u 7 → −∆u, with the Robin boundary conditions ∂u/∂ν = αu at ∂Ω, where ν is the outer unit normal. We show that, for any fixed m ∈ N, the mth eigenvalue EΩm(α) o ..."
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Let Ω ⊂ R2 be the exterior of a convex polygon whose side lengths are `1,..., `M. For a real constant α, let HΩα denote the Laplacian in Ω, u 7 → −∆u, with the Robin boundary conditions ∂u/∂ν = αu at ∂Ω, where ν is the outer unit normal. We show that, for any fixed m ∈ N, the mth eigenvalue EΩm(α) of HΩα behaves as EΩm(α) = −α2 + µDm + O(α−1/2) as α → +∞, where µDm stands for the mth eigenvalue of the operator D1⊕ · · ·⊕DM and Dn denotes the onedimensional Laplacian f 7 → −f ′ ′ on (0, `n) with the Dirichlet boundary conditions.