Some Examples in One–dimensional “Geometric” Scattering

We consider “geometric” scattering for a Laplace-Beltrami operator on a compact Riemannian manifold inserted between “wires,” that is, two half-lines. We discuss applicability and correctness of this model. With an example, we show that such a scattering problem may exhibit unusual properties: the transition coefficient has a sequence of sharp peaks which become more and more distant at high energy and otherwise turns to zero.