Modified wave operators for discrete Schr\"odinger operators with   long-range perturbations
  We consider the scattering theory for discrete Schr\"odinger operators on $Z^d$ with long-range potentials. We prove the existence of modified wave operators constructed in terms of solutions of a Hamilton-Jacobi equation on the torus $T^d$. 
Shu Nakamura
math-ph math.AP math.MP