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

Shihe Liu, Yunfeng Shi, Zhifei Zhang

TL;DR

This work addresses nonlinear Anderson localization for a Maryland quasi-periodic model on $\mathbb{Z}^d$ by diagonalizing the linear Maryland operator and applying a Craig–Wayne–Bourgain (CWB) framework. The authors construct time-quasi-periodic, spatially exponentially decaying localized states for the nonlinear Schrödinger equation $i\partial_t u=Hu+\delta|u|^{2p}u$ in the regime of small $\varepsilon$ and $\delta$ with Diophantine frequency $\alpha$, starting from a finite set of localized linear eigenfunctions. Key contributions include a large deviation theorem for Green's functions across multiple scales, precise eigenvalue separation estimates for the unbounded Maryland potential, and a nonlinear Lyapunov–Schmidt/Nash–Moser scheme that yields quasi-periodic localized solutions with frequencies near the linear eigenvalues. This work extends Anderson localization to nonlinear Maryland-type models in higher dimensions and provides a robust analytical framework that blends spectral analysis with multi-scale probabilistic techniques, delivering both existence and quantitative decay properties of nonlinear localized states. The results have potential implications for understanding interacting quantum systems in quasi-periodic disordered media and for the broader study of nonlinear dynamics in almost-periodic settings.

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.

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

TL;DR

This work addresses nonlinear Anderson localization for a Maryland quasi-periodic model on by diagonalizing the linear Maryland operator and applying a Craig–Wayne–Bourgain (CWB) framework. The authors construct time-quasi-periodic, spatially exponentially decaying localized states for the nonlinear Schrödinger equation in the regime of small and with Diophantine frequency , starting from a finite set of localized linear eigenfunctions. Key contributions include a large deviation theorem for Green's functions across multiple scales, precise eigenvalue separation estimates for the unbounded Maryland potential, and a nonlinear Lyapunov–Schmidt/Nash–Moser scheme that yields quasi-periodic localized solutions with frequencies near the linear eigenvalues. This work extends Anderson localization to nonlinear Maryland-type models in higher dimensions and provides a robust analytical framework that blends spectral analysis with multi-scale probabilistic techniques, delivering both existence and quantitative decay properties of nonlinear localized states. The results have potential implications for understanding interacting quantum systems in quasi-periodic disordered media and for the broader study of nonlinear dynamics in almost-periodic settings.

Abstract

In this paper, we investigate Anderson localization for a nonlinear perturbation of the Maryland model on . 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 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.

Paper Structure

This paper contains 19 sections, 21 theorems, 384 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model and main result
  3. Related results
  4. Main new ingredients of the proof
  5. Structure of the paper
  6. Eigenvalues estimates of the Schrödinger operator
  7. Linear analysis: large deviation theorem (LDT)
  8. Some useful lemmas
  9. LDT in small scales
  10. LDT in intermediate scales
  11. LDT in large scales
  12. Nonlinear analysis: Proof of Theorem \ref{['Main Thm']}
  13. Lyapunov-Schmidt decomposition
  14. The P-equations
  15. The Q-equation
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\alpha\in{\rm DC}_{\gamma,\tau}$ and fix any $b$ distinct lattice sites $\beta_k\in \mathbb{Z}^d,\ k=1,2,\cdots,b$ with $\sup_{k}|\beta_k|\leq B$. Define with $\mathbf{a}=(a_1,\cdots, a_k)\in [1,2]^b$. Then for $0<\delta\leq \tilde{\delta}(\gamma,\tau,b,d,B,p)\ll 1$, there exist a constant $\tilde{\varepsilon}=\tilde{\varepsilon}(\delta)>0$ and a set $\Theta=\Theta_{\alpha,\delta}\subset [0,

Figures (2)

  • Figure 1: The construction of relabelling map in each annulus.
  • Figure 2: Homothetic transform from cubes to vertices.

Theorems & Definitions (35)

  • Theorem 1.1
  • Lemma 2.1: Theorem 1.1, KPS24
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4: Separation property of eigenvalues
  • proof : Proof of Theorem $\ref{['seperative spectrum']}$
  • Lemma 2.5: cf. Lemma A.4 of SW24
  • Lemma 2.6: cf. Lemma A.5 of SW24
  • ...and 25 more