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Wencai Liu

Publications and source records attributed to Wencai Liu.

At least 55 records · Page 3Linked to original sources
arXiv2018

Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators

For perturbed Stark operators $Hu=-u^{\prime\prime}-xu+qu$, the author has proved that $\limsup_{x\to \infty}{x}^{\frac{1}{2}}|q(x)|$ must be larger than $\frac{1}{\sqrt{2}}N^{\frac{1}{2}}$ in order to create $N$ linearly independent eigensolutions in $L^2(\mathbb{R}^+)$. In this paper, we apply generalized Wigner-von Neumann type functions to construct embedded eigenvalues for a class of Schrödinger operators, including a proof that the bound $\frac{1}{\sqrt{2}}N^{\frac{1}{2}}$ is sharp.

math-ph↗
arXiv2018

Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators

In this paper, we consider the perturbed Stark operator \begin{equation*} Hu=H_0u+qu=-u^{\prime\prime}-xu+qu, \end{equation*} where $q$ is the power-decaying perturbation. The criteria for $q$ such that $H=H_0+q$ has at most one eigenvalue (finitely many, infinitely many eigenvalues) are obtained. All the results are quantitative and are generalized to the perturbed Stark type operator.

math-ph↗
arXiv2018

Almost Mathieu operators with completely resonant phases

Let $α\in \mathbb{R}\backslash \mathbb{Q}$ and $β(α) = \limsup _{n \to \infty}(\ln q_{n+1})/ q_n <\infty$, where $p_n/q_n$ is the continued fraction approximations to $α$. Let $(H_{λ,α,θ}u) (n)=u(n+1)+u(n-1)+ 2λ\cos2π(θ+nα)u(n)$ be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $λ, θ\in \mathbb{R}$. Avila and Jitomirskaya \cite{avila2009ten} conjectured that for $2θ\in α\mathbb{Z} + \mathbb{Z}$, $H_{λ,α,θ}$ satisfies Anderson localization if $|λ|>e^{2β(α)}$. In this paper, we developed a method to treat simultaneous frequency and phase resonances and obtain that for $2θ\in α\mathbb{Z}+\mathbb{Z}$, $H_{λ,α,θ}$ satisfies Anderson localization if $|λ|>e^{3β(α)}$.

math.SP↗
arXiv2018

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic Jacobi operators

We are interested in diagonal perturbations of a periodic Jacobi operator that introduce embedded eigenvalues in its essential spectrum. Embedding multiple points in the essential spectrum has been known to be difficult, given that eigenvalues are destroyed easily by small perturbations. However, given a finite or countably infinite set of points within an absolutely continuous band of the original periodic operator (subject only to a very weak non-resonance condition) we are able to construct a diagonal perturbation that preserves the essential spectrum and places eigenvalues in all of those points.

math.SP↗
arXiv2018

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

In this paper, we consider the Schrödinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where $V_0(x)$ is 1-periodic and $V (x)$ is a decaying perturbation. By Floquet theory, the spectrum of $H_0=-\nabla^2+V_0$ is purely absolutely continuous and consists of a union of closed intervals (often referred to as spectral bands). Given any finite set of points $\{ E_j\}_{j=1}^N$ in any spectral band of $H_0$ obeying a mild non-resonance condition, we construct smooth functions $V(x)=\frac{O(1)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$. Given any countable set of points $\{ E_j\}$ in any spectral band of $H_0$ obeying the same non-resonance condition, and any function $h(x)>0$ going to infinity arbitrarily slowly, we construct smooth functions $|V(x)|\leq \frac{h(x)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$. On the other hand, we show that there is no eigenvalue of $H=H_0+V$ embedded in the spectral bands if $V(x)=\frac{o(1)}{1+|x|}$ as $x$ goes to infinity. We prove also an analogous result for Jacobi operators.

math.SP↗
arXiv2018

The asymptotical behaviour of embedded eigenvalues for perturbed periodic operators

Let $H_0$ be a periodic operator on $\R^+$(or periodic Jacobi operator on $\N$). It is known that the absolutely continuous spectrum of $H_0$ is consisted of spectral bands $\cup[α_l,β_l]$. Under the assumption that $\limsup_{x\to \infty} x|V(x)|<\infty$ ($\limsup_{n\to \infty} n|V(n)|<\infty$), the asymptotical behaviour of embedded eigenvalues approaching to the spectral boundary is established.

math-ph↗
arXiv2018

Sharp bound on the largest positive eigenvalue for one-dimensional Schrödinger operators

Let $H=-D^2+V$ be a Schrödinger operator on $ L^2(\mathbb{R})$, or on $ L^2(0,\infty)$. Suppose the potential satisfies $\limsup_{x\to \infty}|xV(x)|=a<\infty$. We prove that $H$ admits no eigenvalue larger than $ \frac{4a^2}{π^2}$. For any positive $a$ and $λ$ with $0<λ< \frac{4a^2}{π^2}$, we construct potentials $V$ such that $\limsup_{x\to \infty}|xV(x)|=a $ and the associated Schrodinger operator $H=-D^2+V$ has eigenvalue $λ$.

math-ph↗
arXiv2018

Proof of the HRT conjecture for almost every (1,3) configuration

Based on the proof of Demeter \We prove that for almost every (1,3) configuration, there is no linear dependence between the associated time-frequency translates of any $f\in L^2(\R)\backslash \{0\}$.cite{Dem10}, we establish the HRT conjecture for almost every (1,3) configuration.

math.FA↗
arXiv2018

Some refined results on mixed Littlewood conjecture for pseudo-absolute values

In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo absolute value sequence $\mathcal{D}$, we obtain the sharp criterion such that for almost every $α$ the inequality \begin{equation*} |n|_{\mathcal{D}}|nα-p|\leq ψ(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$ for a certain one-parameter family of $ψ$. Also under minor condition on pseudo absolute value sequences $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$, we obtain a sharp criterion on general sequence $ψ(n)$ such that for almost every $α$ the inequality \begin{equation*} |n|_{\mathcal{D}_1}|n|_{\mathcal{D}_2}\cdots |n|_{\mathcal{D}_k}|nα-p|\leq ψ(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$.

math.NT↗
arXiv2018

Continuous quasiperiodic Schrödinger operators with Gordon type potentials

Let us concern the quasi-periodic Schrödinger operator in the continuous case, \begin{equation*} (Hy)(x)=-y^{\prime\prime}(x)+V(x,ωx)y(x), \end{equation*} where $V:(\R/\Z)^2\to \R$ is piecewisely $γ$-Hölder continuous with respect to the second variable. Let $L(E)$ be the Lyapunov exponent of $Hy=Ey$. Define $β(ω)$ as \begin{equation*} β(ω)= \limsup_{k\to \infty}\frac{-\ln ||kω||}{k}. \end{equation*} We prove that $H$ admits no eigenvalue in regime $\{E\in\R:L(E)<γβ(ω)\}$.

math-ph↗
arXiv2018

Criteria for embedded eigenvalues for discrete Schrödinger operators

In this paper, we consider discrete Schrödinger operators of the form, \begin{equation*} (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n). \end{equation*} We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$. For $H_0$ (no perturbation), $σ_{\rm ess}(H_0)=σ_{\rm ac}(H)=[-2,2]$ and $H_0$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the {\it almost sign type potential } and develop the Prüfer transformation to address this problem, which leads to the following five results. \begin{description} \item[1] We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominator. \item[2] Suppose $\limsup_{n\to \infty} n|V(n)|=a<\infty.$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator. \item[3] We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$. \item [4]Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(-2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$. \item[5]Given any countable set of points $\{ E_j\}$ in $(-2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct potential $|V(n)|\leq \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$. \end{description}

math.SP↗