Graph-like models for thin waveguides with Robin boundary conditions

We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that in the limit of small width of the waveguide the transverse modes are independent. The projections on each transverse mode generically give rise to decoupling between the edges of the graph while exceptionally a coupling can occur. The coupling takes place if there exists a resonance at the threshold of the continuum spectrum of the effective Hamiltonian resulting from the projection.