Anderson localized states for the nonlinear Maryland model on $\mathbb{Z}^d$

TL;DR

This paper demonstrates the existence of Anderson localized states in a nonlinear Maryland model on bZ^d, using eigenvalue estimates and KAM theory to handle small perturbations and nonlinearities.

Contribution

It introduces a novel approach combining eigenvalue estimates with KAM techniques to establish localization in a nonlinear, quasi-periodic setting.

Findings

Existence of exponentially decaying localized states for small perturbations.

Construction of a large number of quasi-periodic solutions.

Application of KAM theory to nonlinear lattice models.

Abstract

In this paper, we investigate Anderson localization for a nonlinear perturbation of the Maryland model $H=\varepsilon\Delta+\cot\pi(\theta+j\cdot\alpha)\delta_{j,j'}$ on $\mathbb{Z}^d$. Specifically, if $\varepsilon,\delta$ are sufficiently small, we construct a large number of time quasi-periodic and space exponentially decaying solutions (i.e., Anderson localized states) for the equation $i\frac{\partial u}{\partial t}=Hu+\delta|u|^{2p}u$ with a Diophantine $\alpha$. Our proof combines eigenvalue estimates of the Maryland model with the Craig-Wayne-Bourgain method, which originates from KAM theory for Hamiltonian PDEs.