  We prove that the Dirichlet-to-Neumann operator (DtN) has no spectrum in the
lower half of the complex plane. We find several application of this fact in
scattering by obstacles with impedance boundary conditions. In particular, we
find an upper bound for the gradient of the scattering amplitude and for the
total cross section. We justify numerical approximations by providing bounds on
difference between theoretical and approximated solutions without using any a
priory unknown constants.
