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The quantum-mechanical scattering on a compact Riemannian manifold withsemi-axes attached to it (hedgehog-shaped manifold) is considered. The completedescription of the spectral structure of Schroedinger operators on such amanifold is done, the proof of existence and uniqueness of scattering states ispresented, an explicit form for the scattering matrix is obtained and unitarynature of this matrix is proven. It is shown that the positive part of thespectrum of the Schroedinger operator on the initial compact manifold as wellas the spectrum of a point perturbation of such an operator may be recoveredfrom the scattering amplitude for one attached half-line. Moreover, thepositive part of the spectrum of the initial Schroedinger operator is fullydetermined by the conductance properties of an "electronic device" consistingof the initial manifold and two "wires" attached to it

Scattering on compact manifolds with infinitely thin horns