Scattering and inverse scattering for nonlinear quantum walks

TL;DR

This paper investigates the long-term behavior of nonlinear quantum walks, demonstrating scattering phenomena and developing inverse scattering formulas using techniques adapted from nonlinear Schrödinger equations.

Contribution

It introduces the first application of dispersive and Strichartz estimates to nonlinear quantum walks for analyzing their scattering behavior.

Findings

Proves scattering for nonlinear quantum walks in the weak nonlinear regime.

Derives a reproducing formula for inverse scattering in this context.

Adapts methods from nonlinear Schrödinger equations to quantum walks.

Abstract

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schr\"odinger equations and discrete nonlinear Schr\"odinger equations but it seems to be the first time to be applied to QW.