Modified wave operators for discrete Schr\"odinger operators with long-range perturbations

TL;DR

This paper develops a scattering theory for discrete Schrödinger operators with long-range potentials on integer lattices, establishing the existence of modified wave operators via Hamilton-Jacobi equations on the torus.

Contribution

It introduces a novel approach to constructing modified wave operators for discrete Schrödinger operators with long-range perturbations using Hamilton-Jacobi equations.

Findings

Existence of modified wave operators proven.

Construction of wave operators via Hamilton-Jacobi solutions.

Applicable to long-range potentials on Z^d.

Abstract

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 $T^d$.