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.