Some Examples in One-Dimensional ‘‘Geometric’’ Scattering on Manifolds

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