Anderson localized states for the nonlinear Maryland model on $\mathbb{Z}^d$
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913706577756160 |
|---|---|
| author | Liu, Shihe Shi, Yunfeng Zhang, Zhifei |
| author_facet | Liu, Shihe Shi, Yunfeng Zhang, Zhifei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anderson localized states for the nonlinear Maryland model on $\mathbb{Z}^d$ Liu, Shihe Shi, Yunfeng Zhang, Zhifei Analysis of PDEs Mathematical Physics Dynamical Systems Spectral Theory 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. |
| title | Anderson localized states for the nonlinear Maryland model on $\mathbb{Z}^d$ |
| topic | Analysis of PDEs Mathematical Physics Dynamical Systems Spectral Theory |
| url | https://arxiv.org/abs/2502.16397 |