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

TL;DR

This paper develops Isozaki-Kitada modifiers for discrete Schrödinger operators on lattices, establishing the existence and completeness of modified wave operators for long-range potentials on periodic graphs.

Contribution

It constructs time-independent Isozaki-Kitada modifiers for discrete Schrödinger operators on general lattices, extending scattering theory methods.

Findings

Existence of modified wave operators with Isozaki-Kitada modifiers.

Completeness of the constructed wave operators.

Application to discrete Schrödinger operators on periodic graphs.

Abstract

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