  The purpose of the present paper is to develop the inverse scattering
transform for the nonlocal semi-discrete nonlinear Schrodinger equation (known
as Ablowitz-Ladik equation) with PT-symmetry. This includes: the eigenfunctions
(Jost solutions) of the associated Lax pair, the scattering data and the
fundamental analytic solutions. In addition, the paper studies the spectral
properties of the associated discrete Lax operator. Based on the formulated
(additive) Riemann-Hilbert problem, the 1- and 2-soliton solutions for the
nonlocal Ablowitz-Ladik equation are derived. Finally, the completeness
relation for the associated Jost solutions is proved. Based on this, the
expansion formula over the complete set of Jost solutions is derived. This will
allow one to interpret the inverse scattering transform as a generalised
Fourier transform.
