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This is a Quantitative Study study.

October 7, 2025Open Access

Anderson localized states for the nonlinear Maryland model on Zᵈ

Key Points

Numerous Anderson localized states were constructed under specific conditions of small perturbations.
The research highlights the relationship between eigenvalue estimates and nonlinear dynamics, revealing significant outcomes.
Using the Craig-Wayne-Bourgain method alongside the Maryland model, the study successfully demonstrates localization properties.
The findings suggest potential advancements in understanding nonlinear perturbations in quantum systems.

Abstract

In this paper, we investigate Anderson localization for a nonlinear perturbation of the Maryland model H=+ (+j) ₉, ₉' on Zᵈ. Specifically, if, 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 u t=Hu+|u|^2pu with a Diophantine. Our proof combines eigenvalue estimates of the Maryland model with the Craig-Wayne-Bourgain method, which originates from KAM theory for Hamiltonian PDEs.

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