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Wencai Liu

Publications and source records attributed to Wencai Liu.

66 records · Page 4Linked to original sources
arXiv2018

Growth of the eigensolutions of Laplacians on Riemannian manifolds I: construction of energy function

In this paper, we consider the eigen-solutions of $-Δu+ Vu=λu$, where $Δ$ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato's methods on manifold and establish the growth of the eigen-solutions as $r$ goes to infinity based on the asymptotical behaviors of $Δr$ and $V(x)$, where $r=r(x)$ is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies $ K_{\rm rad}(r)= -1+\frac{o(1)}{r}$.

math.DG↗
arXiv2018

Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.

math-ph↗
arXiv2017

Universal hierarchical structure of quasiperiodic eigenfunctions

We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.

math-ph↗
arXiv2016

Arithmetic Spectral Transitions for the Maryland Model

We give a precise description of spectra of the Maryland model $ (h_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+ λ\tan π(θ+nα)u_n$ for all values of parameters. We introduce an arithmetically defined index $δ(α, θ)$ and show that for $α\notin\mathbb{Q},\,$ $σ_{sc}(h_{λ,α,θ})=\overline{\{e:γ_λ(e) <δ(α, θ) \}}$ and $σ_{pp}(h_{λ,α,θ})=\{e:γ_λ(e) \geq δ(α, θ) \}$. Since $σ_{ac}(h_{λ,α,θ})=\emptyset,\;$ this gives complete description of the spectral decomposition for {\it all} values of parameters $λ,α,θ$, making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion. The set of eigenvalues can be explicitly identified for all parameters, using the {\it quantization condition}. We also establish, for the first time for this or any other model, a quantization condition for singular continuous spectrum (an arithmetically defined measure zero set that supports singular continuous measures) for all parameters.

math-ph↗
arXiv2016

The spectra of surface Maryland model for all phases

We study the discrete Schrödinger operators $H_{λ,α,θ}$ on $\ell^2(\mathbb{Z}^{d+1})$ with surface potential of the form $V(n,x)=λδ(x)\tanπ(α\cdot n+θ)$, and $H_{λ,α,θ}^{+}$ on $\ell^2(\mathbb{Z}^{d}\times \mathbb{Z}_+)$ with the boundary condition $ ψ_{(n,-1)}=λ\tanπ(α\cdot n+θ)ψ_{(n,0)} $, where $α\in \mathbb{R}^d$ is rationally independent. We show that the spectra of $H_{λ,α,θ}$ and $H_{λ,α,θ}^{+}$ are $(-\infty,\infty)$ for all parameters. We can also determine the absolutely continuous spectra and Hausdorff dimension of the spectral measures if $d=1$.

math-ph↗
arXiv2013

Absolutely Continuous Spectrum for the Quasi-periodic Schrödinger Operator in Exponential Regime

Avila and Jitomirskaya prove that the quasi-periodic Schrödinger operator $H_{λv,α,θ}$ has purely absolutely continuous spectrum for $α$ in sub-exponential regime (i.e., $β(α)=0$) with small $λ$, if $v$ is real analytic in a strip of real axis. In the present paper, we show that for all $α$ with $0<β(α)<\infty$, $H_{λv,α,θ}$ has purely absolutely continuous spectrum with small $λ$, if $v$ is real analytic in strip $ |\Im x|< Cβ$, where $C$ is a large absolute constant.

math.SP↗
arXiv2013

Anderson localization for the completely resonant phases

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}$.

math.SP↗
arXiv2013

Hölder Continuity of the Spectral Measures for One-Dimensional Schrödinger Operator in Exponential Regime

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.

math.ST↗
arXiv2013

Spectral Gaps of Almost Mathieu Operator in Exponential Regime

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 $λ$.

math.SP↗
arXiv2013

Anderson Localization for the Almost Mathieu Operator in Exponential Regime

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}$.

math.SP↗