  We study discrete Schroedinger operators with compactly supported potentials
on the square lattice. Constructing spectral representations and representing
S-matrices by the generalized eigenfunctions, we show that the potential is
uniquely reconstructed from the S-matrix of all energies. We also study the
spectral shift function for the trace class potentials, and estimate the
discrete spectrum in terms of the moments of the spectral shift function and
the potential.
