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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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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.
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 ..."
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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 ..."
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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.
Quantum networks modelled by graphs
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"... 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 ..."
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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.
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"... Abstract. We prove some commutation relations for a 3graded Lie algebra, i.e., a Zgraded Lie algebra whose nonzero homogeneous elements have degrees −1, 0 or 1, over a field K. In particular, we examine the free 3graded Lie algebra generated by an element of degree −1 and another of degree 1. We ..."
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Abstract. We prove some commutation relations for a 3graded Lie algebra, i.e., a Zgraded Lie algebra whose nonzero homogeneous elements have degrees −1, 0 or 1, over a field K. In particular, we examine the free 3graded Lie algebra generated by an element of degree −1 and another of degree 1. We show that if K has characteristic zero, such a Lie algebra can be realized as a Lie algebra of matrices over polynomials in one indeterminate. In the end, we apply the results obtained to derive the classical commutation relations for elements in the universal enveloping algebra of sl2(K).
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"... Abstract. 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 s ..."
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Abstract. 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
QUANTUM NETWORKS MODELLED BY GRAPHS
"... 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 ..."
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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. 1.