J. Bruening and V. Geyler
Scattering on compact manifolds with infinitely thin horns
(143K, LaTeX)

ABSTRACT.  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.