  The quantum-mechanical scattering on a compact Riemannian manifold with
semi-axes attached to it (hedgehog-shaped manifold) is considered. The complete
description of the spectral structure of Schroedinger operators on such a
manifold is done, the proof of existence and uniqueness of scattering states is
presented, an explicit form for the scattering matrix is obtained and unitary
nature of this matrix is proven. It is shown that the positive part of the
spectrum of the Schroedinger operator on the initial compact manifold as well
as the spectrum of a point perturbation of such an operator may be recovered
from the scattering amplitude for one attached half-line. Moreover, the
positive part of the spectrum of the initial Schroedinger operator is fully
determined by the conductance properties of an "electronic device" consisting
of the initial manifold and two "wires" attached to it.
