Many quantum graphs with the same geometry

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Quantum graph models are both interesting mathematically and useful in applications. One of the important questions they pose concerns the meaning of the vertex coupling. The self-adjointness requirement alone leaves a substantial freedom expressed through parameters appearing in the conditions matching the wave function at the graph vertices. It is a longstanding problem whether one can motivate their choice by approximating the graph Hamiltonian by operators on a family of networks, i.e. systems of tubular manifolds the transverse size of which tends to zero. It appears that the answer depends on the conditions imposed on tube boundaries. In this talk we present a complete solution for Neumann networks: we demonstrate that adding properly scaled potentials and changing locally the graph topology, one can approximate any admissible vertex coupling. The result comes from a common work with Taksu Cheon, Olaf Post, and Ondrej Turek.

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