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 Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d$$\end{document}, including the discrete Schrödinger operators on the square lattice. The modifiers are constructed by observing the corresponding Hamilton flow on T∗Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*\mathbb {T}^d$$\end{document}. We prove the existence and completeness of modified wave operators in terms of the above-mentioned time-independent modifiers.