We study discrete Schr?dinger operators with compactly supported potentials on Z ~d. 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 ξ(λ) for the trace class potentials, and estimate the discrete spectrum in terms of the moments of ξ(λ) and the potential.
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