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We consider a scattering theory for convolution operators on$\mathcal{H}=\ell^2(\mathbb{Z}^d; \mathbb{C}^n)$ perturbed with a long-rangepotential $V:\mathbb{Z}^d\to\mathbb{R}^n$. One of the motivating examples isdiscrete Schr\"odinger operators on $\mathbb{Z}^d$-periodic graphs. Weconstruct time-independent modifiers, so-called Isozaki-Kitada modifiers, andwe prove that the modified wave operators with the above-mentionedIsozaki-Kitada modifiers exist and that they are complete.

Construction of Isozaki-Kitada modifiers for discrete Schr\"odinger operators on general lattices