Some examples in one-dimensional ``geometric'' scattering on manifolds (Q1364842)

Let \(\Omega\) be a compact Riemannian manifold which is inserted between two half lines; this means that one glues on two copies of \([0,\infty)\) to two points \(x_1\) and \(x_2\) of a compact Riemannian manifold \(M\). The standard Laplacian \(\Delta_M\) is chosen on \(M\); the operator \(-\partial/\partial x^2\) is chosen on the two half lines. Boundary conditions are chosen at the gluing points \(x_i\) so that the Laplacian does not decompose as a direct sum with summands acting on the different geometric components. The spectrum of the resulting operator is studied as are the transition coefficient in the scattering problem. The author applies these results to the particular example in which \(M\) is the unit sphere in \(\mathbb{R}^3\) and in which the two half lines are glued on at antipodal points; spherical harmonics are used to analyze the situation and the large energy setting is studied.