  The propagator of the discrete Schr\"odinger equation is computed and its
properties are revealed through a Feynman path summation in discrete space.
Initial data problems such as diffraction in discrete space and continuous time
are studied analytically by the application of the new propagator. In the
second part of this paper, the analogy between time propagation and 2D
scattering by 1D obstacles is explored. New results are given in the context of
diffraction by edges within a periodic medium. A connection with tight-binding
arrays and photonic crystals is indicated.
