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We establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on $\mathbb{Z}^d$ with power-law long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic version of localization, the finite volume version of $(\frac12-)$-H\"older continuity of the IDS, and the absence of eigenvalues (for Aubry dual operators)., Comment: Comments welcome; 67 pages

We establish the Anderson localization and exponential dynamical localization for a class of quasi-periodic Schr\"odinger operators on $\mathbb{Z}^d$ with bounded or unbounded Lipschitz monotone potentials via multi-scale analysis based on Rellich function analysis in the perturbative regime. We show that at each scale, the resonant Rellich function uniformly inherits the Lipschitz monotonicity property of the potential via a novel Schur complement argument., Comment: Comments welcome; 62 pages

We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schr\"odinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [GAFA 17(3), 682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [JEMS 10(1), 1--45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma in [GAFA 17(3), 682--706, 2007]., Comment: Comments welcome, 49 pages. arXiv admin note: substantial text overlap with arXiv:2306.00513

In this paper, we establish the Anderson localization, strong dynamical localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for some multi-dimensional discrete quasi-periodic (QP) Schr\"odinger operators with asymmetric $C^2$-cosine type potentials. We extend both the iteration scheme of \cite{CSZ23a} and the interlacing method of \cite{FV21} to handle asymmetric Rellich functions with collapsed gaps., Comment: Comments welcome. 76 pages, the latest version

Using a Birkhoff normal form transform to impede mode transfer in a finite "barrier", we prove localization of arbitrary $\ell^2$ data for polynomially long time for the nonlinear quasi-periodic Schr\"odinger equation on $\mathbb Z^d$., Comment: Comments welcome

We establish Diophantine type estimates on shifts of trigonometric polynomials on the torus $\mathbb{T}^d$, as well as that of their square roots. These estimates arise from the spectral analysis of the quasi-periodic Schr\"odinger and the quasi-periodic wave operators. They have applications to the nonlinear quasi-periodic Schr\"odinger equations (NLS) and the nonlinear quasi-periodic wave equations (NLW). One could now, for example, extend the result of Bourgain (Geom. Funct. Anal. 17(3): 682-706, 2007) to the nonlinear setting., Comment: It is combined in arXiv:2405.17513 and is not intended to be published

We establish large sets of Anderson localized states for the quasi-periodic nonlinear wave equation on $\mathbb Z^d$, thus extending nonlinear Anderson localization from the random \cite{BW08, LW22, GSW23} to a deterministic setting., Comment: 37 pages

In this paper we consider the localization properties of coupled harmonic oscillators in random media. Each of these oscillators is restricted to the lattice $\mathbb{Z}^d$. We show that for most states and an arbitrary choice of the random media, the long time localization for the coupled system holds in a time scale larger than the polynomial one., Comment: 17 pages, comments welcome

In this paper, we study the multidimensional lattice Schr\"odinger operators with $C^2$-cosine like quasi-periodic (QP) potential. We establish quantitative Green's function estimates, the arithmetic version of Anderson (and dynamical) localization, and the finite volume version of $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. Our proof is based on an extension of the fundamental multi-scale analysis (MSA) type method of Fr\"ohlich-Spencer-Wittwer [\textit{Comm. Math. Phys.} 132 (1990): 5--25] to the higher lattice dimensions. We resolve the level crossing issue on eigenvalues parameterizations in the case of both higher lattice dimension and $C^2$ regular potential., Comment: 63 pages, to appear in CMP

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}., Comment: 52 pages, a revised version

In this paper we develop a Nash-Moser iteration type reducibility approach to prove the (inverse) localization for some $d$-dimensional discrete almost-periodic operators with power-law long-range hopping. We also provide a quantitative lower bound on the regularity of the hopping. As an application, some results of \cite{Sar82, Pos83, Cra83, BLS83} are generalized to the power-law hopping case., Comment: to appear in CMP

In this paper, we prove the power-law in time upper bound for the diffusion of a 1D discrete nonlinear Anderson model. We remove completely the decaying condition restricted on the nonlinearity of Bourgain-Wang (Ann. of Math. Stud. 163: 21--42, 2007.). This gives a resolution to the problem of Bourgain (Illinois J. Math. 50: 183--188, 2006.) on diffusion bound for nonlinear disordered systems. The proof uses a novel ``norm'' based on tame property of the Hamiltonian., Comment: While our Birkhoff normal form theory still works (from page 5 to page 17), we find the derivatives estimates on page 19 can not be obtained without any prior estimates of type H^a for small a>0

In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schr\"odinger operator $H=-\Delta+ \lambda V(x),$ where $V$ is a real analytic quasi-periodic function and $\lambda>0$. We first show that $H$ has no eigenvalues in \textit{low energy region}. We also provide in \textit{low energy region} the new phase transition parameter, i.e. the competition between the strength of coupling and the length for frequencies., Comment: Comments welcome, a revised version, 22 pages

In this paper, we confirm the conjecture of Wang and Zhang (J. Stat. Phys. 134 (5-6): 953--968, 2009) in a long time scale, i.e., the displacement of the wavefront for $1D$ nonlinear random Schroedinger equation is of logarithmic order in time $|t|$., Comment: A revised version, to appear in J. Stat. Phys

In this paper we give a multi-scale analysis proof of \textit{power-law} localization for random operators on ${\Z}^d$ for \textit{arbitrary} $d\geq1$., Comment: A revised version. 21 pages

In this paper, we prove the Melnikov's persistency theorem by combining the traditional Kolmogorov-Arnold-Moser (KAM) technique and the Craig-Wayne-Bourgain (CWB) method. The aim of this paper is twofold. One is to establish the linear stability of the perturbed invariant tori by using the CWB method without the second Melnikov condition. The other one is to illustrate the CWB method in detail and make the CWB method more accessible.

In this paper, we study the following nonlinear Schr\"odinger equation \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u=0,\ x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}, \end{eqnarray} where $V*$ is the Fourier multiplier defined by $\widehat{(V* u})_n=V_{n}\widehat{u}_n, V_n\in[-1,1]$ and $f(x)$ is Gevrey smooth. It is shown that for $0\leq|\epsilon|\ll1$, there is some $(V_n)_{n\in\mathbb{Z}}$ such that, the equation admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain \cite{BJFA2005} and Cong-Liu-Shi-Yuan \cite{CLSY} to the case that the nonlinear perturbation depends explicitly on the space variable $x$. The main difficulty here is the absence of zero momentum of the equation.

In this paper, we consider the nonlinear elliptic equations on rectangular tori. Using methods in the study of KAM theory and Anderson localization, we prove that these equations admit many analytic solutions., Comment: Comments are welcome

In this paper, we study the quasi-periodic operators $H_{\epsilon,\omega}(x)$: $$(H_{\epsilon,\omega}(x)\vec{\psi})_n=\epsilon\sum_{k\in\mathbb{Z}}W_k\vec{\psi}_{n-k}+V(x+n\omega)\vec{\psi}_n,$$ where $$\vec{\psi}=\{\vec{\psi}_n\}\in\ell^2(\mathbb{Z},\mathbb{C}^l),\ V(x)=\text{diag}\left(v_1(x),\cdots,v_l(x)\right)$$ with $v_i$ ($1\leq i \leq l$) being real analytic functions on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $W_k$ ($k\in\mathbb{Z}$) being $l\times l$ matrices satisfying $\|W_k\|\leq C_0e^{-\rho|k|}$. Using techniques developed by Bourgain and Goldstein [\textit{{Ann. of Math. 152(3):835--879, 2000}}], we show that for $|\epsilon|\leq \epsilon_{0}(V,\rho,l,C_0)$ ( depending only on $V,\rho, l, C_0$) and $x\in \mathbb{R}/\mathbb{Z}$, there is some full Lebesgue measure subset $\mathcal{F}$ of the Diophantine frequencies such that $H_{\epsilon,\omega}(x)$ exhibits Anderson localization if $\omega\in \mathcal{F}$., Comment: a revised version

In this paper, it is proved that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., \textbf{229} (2005), no. 1, 62-94.] is stable in a very long time for 1D nonlinear Schr\"{o}dinger equation with periodic boundary conditions.