Affiliations: Doppler Institute, Czech Technical University, Prague, Czech Republic. E-mail: [email protected] | Department of Mathematics “G. Castelnuovo”, University of Rome “La Sapienza”, P. le A. Moro 2, 00185 Rome, Italy. E-mail: [email protected]
Note: [] Current address: Institut für Angewandte Mathematik, Endenicher Allee 60, 53115 Bonn, Germany.
Abstract: 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.
