Asymptotic stability of small bound state of nonlinear quantum walks

TL;DR

This paper investigates the long-term behavior of small initial data in nonlinear quantum walks, demonstrating the decomposition into bound states and scattering waves when the linear operator has two eigenvalues.

Contribution

It provides a detailed analysis of the asymptotic stability and decomposition of solutions in nonlinear quantum walks with specific spectral properties.

Findings

Solutions decompose into bound states and scattering waves.

Long-term stability of small bound states is established.

Analysis applies to quantum walks with exactly two eigenvalues.

Abstract

In this paper, we study the long time behavior of nonlinear quantum walks when the initial data is small in $l^2$. In particular, we study the case where the linear part of the quantum walk evolution operator has exactly two eigenvalues and show that the solution decomposed into nonlinear bound states bifurcating from the eigenvalues and scattering waves.