Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds

Details Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds

2008-11-22

arxiv.org/abs/0811.3707v1

Physics

Mathematical Physics

19 pages, 1 figure

Scientific paper

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 Schr\"odinger operators can approximate non-trivial vertex couplings. The latter include not only the delta-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric delta'-couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal.

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