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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.
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.