Vertex couplings in quantum graphs (量子科学における双対性とスケール)

概要

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We propose a parametrization of the family of all physically admissible couplings in quantum graph vertices. With the help of this result we solve the longstanding open problem of approximating by regular graphs all singular vertex couplings in quantum graph vertices. The approximating graph is constructed as follows. We begin with a star graph with n arms and a general coupling in the vertex - the approximated system. Then we consider a system of n half lines whose endpoints are connected by lines of a length d such that there are δ couplings and δ interactions placed in the endpoints of the half lines and in the centers of the connecting lines; at the same time, the connecting lines are supposed to support vector potentials - the approximating system. It turns out that the approximated system can be considered as a limit of the family of approximating systems as d → 0, where the convergence is in the norm-resolvent sense.
2012-02-20