Long-Range Scattering for Discrete Schrödinger Operators

In this paper, we define time-independent modifiers to construct a long-range scattering theory for a class of difference operators on \(\mathbb {Z}^d\), including the discrete Schrodinger operators on the square lattice. The modifiers are constructed by observing the corresponding Hamilton flow on \(T^*\mathbb {T}^d\). We prove the existence and completeness of modified wave operators in terms of the above-mentioned time-independent modifiers.