arXiv subjects
Publications and source records attributed to Xiaoping Yuan.
For the almost Mathieu operator $ (H_{λ,α,θ}u) (n)=u(n+1)+u(n-1)+ λv(θ+nα)u(n)$, Avila and Jitomirskaya guess that for every phase $ θ\in \mathscr{R} \triangleq\{θ\in \mathbb{R}\;| \; 2θ+ α\mathbb{Z} \in \mathbb{Z}\}$, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ 2 β}$. In the present paper, we show that for every phase $ θ\in \mathscr{R} $, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ 7 β}$.
Avila and Jitomirskaya prove that the spectral measure $μ_{λv, α,x}^f$ of quasi-periodic Schrödinger operator is $1/2$-Hölder continuous with appropriate initial vector $f$, if $α$ satisfies Diophantine condition and $λ$ is small. In the present paper, the conclusion is extended to that for all $α$ with $β(α)<\infty$, the spectral measure $μ_{λv, α,x}^f$ is $1/2$-Hölder continuous with small $λ$, if $v$ is real analytic in a neighbor of $\{|\Im x|\leq Cβ\}$, where $C$ is a large absolute constant. In particular, the spectral measure $μ_{λ, α,x}^f$ of almost Mathieu operator is $1/2$-Hölder continuous if $|λ|<e^{-Cβ}$ with $C$ a large absolute constant.
For almost Mathieu operator $(H_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+2λ\cos2π(θ+nα)u_n$, the dry version of Ten Martini problem predicts that the spectrum $Σ_{λ,α}$ of $ H_{λ,α,θ}$ has all gaps open for all $λ\neq 0$ and $ α\in \mathbb{R}\backslash \mathbb{Q}$. Avila and Jitomirskaya prove that $Σ_{λ,α}$ has all gaps open for Diophantine $α$ and $0<|λ|<1$. In the present paper, we show that $Σ_{λ,α}$ has all gaps open for all $ α\in \mathbb{R}\backslash \mathbb{Q}$ with small $λ$.
For the almost Mathieu operator $(H_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+2λ\cos2π(θ+nα)u_n$, Avila and Jitomirskaya guess that for a.e. $θ$, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ β} $, and they establish this for $ |λ| > e^{\frac{16}{9} β}$. In the present paper, we extend their result to regime $ |λ| > e^{\frac{3}{2} β}$.
Introduce several KAM theorems for infinite dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori. Especially, introduce a KAM theorem in the paper(Cummun. Math. Phys. 226, 61-100 (2002)) which shows that there are rich KAM tori for a class of Hamiltonian with short range and with linearized operator of pure point spectra. Here are also presented several open problems.
The existence of lower dimensional KAM tori is shown for a class of nearly integrable Hamiltonian systems where the second Melnikov's conditions are eliminated. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for nonlinear wave equations, Schrödinger equations and other equations of higher spatial dimension.