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Wenwen Jian

Publications and source records attributed to Wenwen Jian.

8 recordsLinked to original sources
arXiv2026

Almost-sure quenched KAM tori for cubic NLS with a spatial white-noise potential

Let \[ A_ω=-\partial_x^2+ρ\dot B_x(ω), \qquad ρ\ne0, \] be the Dirichlet Schrödinger operator on $(0,π)$ with a spatial Gaussian white-noise potential, realized pathwise via quasi-derivatives. Here $\dot B_x$ denotes the distributional derivative of Brownian motion with respect to the spatial variable $x$. For $κ\ne0$, we consider the cubic nonlinear Schrödinger equation \[ i u_t=A_ωu+κ|u|^2u. \] We first prove a zero-set theorem for locally real-analytic functions on classical Wiener space. As a consequence, on a single event of probability one, no nontrivial finitely supported integer combination of the random eigenvalues vanishes, and the quartic twist matrix is nonsingular for every finite tangential set. After fixing a path in this event, we combine these qualitative nondegeneracy properties with a partial quartic Birkhoff normal form adapted to the KAM scheme. For every finite nonempty tangential set \(J\) of cardinality \(b\), we obtain a Cantor family of linearly stable, real-analytic, small-amplitude invariant \(b\)-tori. The family is parametrized by a Cantor subset of \([ν,2ν]^b\) whose relative measure tends to one as \(ν\to0\).

math.PR↗
arXiv2026

High-energy asymptotics for finite-interval Schrödinger operators with Gaussian white-noise potential

We study the one-dimensional Schrödinger operator on a fixed interval with Gaussian white-noise potential, \[ H_ω=-\frac{\dd^2}{\dd x^2}+ρ\dot B_x(ω), \] under Dirichlet boundary conditions. The operator is defined pathwise through the quasi-derivative realization of Sturm--Liouville operators with distributional potentials. Let $λ_n$ be the Dirichlet eigenvalues, $λ_n^+=\max\{λ_n,0\}$, and $k_n=\sqrt{λ_n^+}$. For every finite $p$, we prove the high-energy expansion \[ k_n=\frac{nπ}{L} +\fracρ{nπ}\int_0^L \sin^2\left(\frac{nπs}{L}\right)\,\dd B_s +O_{L^p(Ω)}(n^{-2}). \] Consequently, almost surely, $λ_n>0$ for all sufficiently large $n$ and, for every $\varepsilon>0$, \[ k_n=\frac{nπ}{L}+O(n^{-1+\varepsilon}). \] We also obtain first-order eigenfunction asymptotics with explicit Brownian stochastic-integral corrections. In particular, for the $L^2(0,L)$-normalized Dirichlet eigenfunction $φ_n$, with a fixed sign convention, \[ \sup_{0\le x\le L} \left|φ_n(x)-\sqrt{\frac{2}{L}}\sin(k_n x)\right| =O(n^{-1+\varepsilon}) \] almost surely. The proofs use stochastic Prüfer coordinates, stochastic Volterra expansions, the Burkholder--Davis--Gundy inequality, and a Borel--Cantelli argument. The estimates provide a first step toward KAM-type small-divisor analysis for Hamiltonian PDEs with white-noise spatial potentials.

math.SP↗
arXiv2020

Krylov--Bogolyubov averaging

We present the modified approach to the classical Bogolyubov-Krylov averaging, developed recently for the purpose of PDEs. It allows to treat Lipschitz perturbations of linear systems with pure imaginary spectrum and may be generalized to treat PDEs with small nonlinearities.

math.DS↗
arXiv2019

Reducibility of the quantum harmonic oscillator in $d$-dimensions with finitely differentiable perturbations

In this paper, the $d$-dimensional quantum harmonic oscillator with a pseudo-differential time quasi-periodic perturbation \begin{equation}\label{0} \text{i}\dotψ=(-Δ+V(x)+εW(ωt,x,-\text{i}\nabla))ψ,\ \ \ \ \ x\in\mathbb{R}^d \end{equation} is considered, where $ω\in(0,2π)^n$, $V(x):=\sum_{j=1}^d v_j^2x_j^2, v_j\geq v_0>0$, and $W(θ,x,ξ)$ is a real polynomial in $(x,ξ)$ of degree at most two, with coefficients belonging to $C^{\ell}$ in $θ\in\mathbb{T}^n$ for the order $\ell$ satisfying $\ell\geq 2n-1+β,\ 0<β<1$. Using techniques developed by Bambusi-Grébert-Maspero-Robert [\emph{Anal. PDE. 11(3):775-799, 2018}] and Rüssmann [\emph{pages 598--624. Lecture Notes in Phys., Vol. 38, 1975}], the paper shows that for any $|ε|\leq ε_{\star}(n,\ell)$, there is a set $\mathcal{D}_ε\subset (0,2π)^n$ with big Lebesgue measure, such that for any $ω\in\mathcal{D}_ε$, the system is reducible.

math.DS↗
arXiv2018

Anderson localization for one-frequency quasi-periodic block operators with long-range interactions

In this paper, we study the quasi-periodic operators $H_{ε,ω}(x)$: $$(H_{ε,ω}(x)\vecψ)_n=ε\sum_{k\in\mathbb{Z}}W_k\vecψ_{n-k}+V(x+nω)\vecψ_n,$$ where $$\vecψ=\{\vecψ_n\}\in\ell^2(\mathbb{Z},\mathbb{C}^l),\ V(x)=\text{diag}\left(v_1(x),\cdots,v_l(x)\right)$$ 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 $\|W_k\|\leq C_0e^{-ρ|k|}$. Using techniques developed by Bourgain and Goldstein [\textit{Ann. of Math. 152(3):835--879, 2000}], we show that for $|ε|\leq ε_{0}(V,ρ,l,C_0)$ ( depending only on $V,ρ, l, C_0$) and $x\in \mathbb{R}/\mathbb{Z}$, there is some full Lebesgue measure subset $\mathcal{F}$ of the Diophantine frequencies such that $H_{ε,ω}(x)$ exhibits Anderson localization if $ω\in \mathcal{F}$.

math.SP↗