Year: 2025
Type: preprint
Abstract: In this paper, we investigate Anderson localization for a nonlinear perturbation of the Maryland model $H=\varepsilonΔ+\cotπ(θ+j\cdotα)δ_{j,j'}$ on $\mathbb{Z}^d$. Specifically, if $\varepsilon,δ$ 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 $i\frac{\partial u}{\partial t}=Hu+δ|u|^{2p}u$ with a Diophantine $α$. Our proof combines eigenvalue estimates of the Maryland model with the Craig-Wayne-Bourgain method, which originates from KAM theory for Hamiltonian PDEs.
Source: arXiv (Cornell University)
Authors: Liu, Shihe, Shi, Yunfeng, Zhang, Zhifei
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4414840922
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