The Fredholm Analytic Theorem is a standard instrument of the spectral
analysis having numerous applications. Its main field is the perturbation
of the discrete spectra. For the purpouses of the continuous spectra
perturbation ( say in scattering theory) one needs to apply a
version of the Fredholm Analytic Theorem where the invertibility is
investigated on the boundary of a domain ( for example on the real axis in
case of the upper half-plane). We present some ``boundary'' version of the
theorem. Applications to the scattering theory are considered.