Convergence of resonances on thin branched quantum waveguides

We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family Xε of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on Xε approximate those of the Laplacian with “free” boundary conditions on X0, the skeleton graph of Xε.

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