**Abstract: **
A limiting case of the quantum mechanical scattering
on the horned Riemannian manifolds is considered;
namely, we assume the widths of the horns to be
tending to zero. The scattering matrix is obtained in
an explicit form, its unitarity is proved. Some examples
are discussed in detail; in particular, in the case of
Riemann surfaces of constant negative curvature a relation
of the scattering amplitude to the Selberg zeta-function
is explained. In the case of non-compact manifolds,
an analogue of the optical theorem is presented.