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Opérateurs de Schrödinger sur des graphes métriques

Abstract : This thesis is devoted to investigation of quantum graphs, in other words, quantum systems in which a nonrelativistic particle is confined to a graph. We propose a new way to represent the boundary conditions, and with the help of this result we solve the longstanding open problemof approximating by regular graphs all singular vertex couplings in quantum graph vertices. We present a construction in which the edges are disjunct and the pairs of the so obtained endpoints are joined by additional connecting edges of lengths 2d. Each connecting edge carries a delta potential and a vector potential. It is shown that when the lengths 2d of the connecting edges shrink to zero and the added potentials properly depend on d, the limit can yield any requested singular vertex coupling. This type of boundary conditions is used to examine scattering properties of singular vertices of degrees 2 and 3. We show thar the couplings between each pair of the outgoing edges are individually tunable, which could enable the design of quantum spectral junctions filters. We also study Schrödinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by delta-couplings. If the graph is periodic, the Hamiltonian has a band spectrum. We consid a "bending" deformation of the chain consisting in changing the position of the point of contact between two rings. We show that this deformation gives rive to eigenvalues and analyze their dependence on the "bending angle".
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Submitted on : Wednesday, October 20, 2010 - 11:57:27 AM
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  • HAL Id : tel-00527790, version 1



Ondrej Turek. Opérateurs de Schrödinger sur des graphes métriques. Mathématiques [math]. Université du Sud Toulon Var, 2009. Français. ⟨tel-00527790⟩



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