Criteria for embedded eigenvalues for discrete Schr\"odinger operators

In this paper, we consider discrete Schr\"odinger 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), $\sigma_{\rm ess}(H_0)=\sigma_{\rm ac}(H)=[-2,2]$ 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 Pr\"ufer 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

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

In this paper, we consider the eigen-solutions of $-\Delta u+ Vu=\lambda u$, where $\Delta$ 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 $\Delta r$ 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 $K_{\rm rad}(r)= -1+\frac{o(1)}{r}$.Comment: IMRN to appea

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 $\alpha$ the inequality \begin{equation*} |n|_{\mathcal{D}}|n\alpha -p|\leq \psi(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$ for a certain one-parameter family of $\psi$. 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 $\psi(n)$ such that for almost every $\alpha$ the inequality \begin{equation*} |n|_{\mathcal{D}_1}|n|_{\mathcal{D}_2}\cdots |n|_{\mathcal{D}_k}|n\alpha-p|\leq \psi(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$.Comment: J. Aust. Math. Soc. to appea

Growth of the eigensolutions of Laplacians on Riemannian manifolds II: positivity of the initial energy

In this paper, energy function is used to investigate the eigen-solutions of $-\Delta u+ Vu=\lambda u$ on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions

Continuous quasiperiodic Schr\"odinger operators with Gordon type potentials

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

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.Comment: J. Funct. Anal. to appea

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 Schr\"odinger operators, including a proof that the bound $\frac{1}{\sqrt{2}}N^{\frac{1}{2}}$ is sharp

Absence of singular continuous spectrum for perturbed discrete Schr\"odinger operators

We show that the spectral measure of discrete Schr\"odinger operators $(Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$

Sharp bound on the largest positive eigenvalue for one-dimensional Schr\"odinger operators

Let $H=-D^2+V$ be a Schr\"odinger 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 $\frac{4a^2}{\pi^2}$. For any positive $a$ and $\lambda$ with $0<\lambda< \frac{4a^2}{\pi^2}$, we construct potentials $V$ such that $\limsup_{x\to \infty}|xV(x)|=a$ and the associated Sch\"rodinger operator $H=-D^2+V$ has eigenvalue $\lambda$.Comment: After we finished this note, we noticed that the main result has been proved by Halvorsen and Atkinson-Everitt. So this paper is not intended for publicatio

Inhomogeneous Diophantine approximation in the coprime setting

Given $n\in N$ and $x,\gamma\in R$, let \begin{equation*} ||\gamma-nx||^\prime=\min\{|\gamma-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number $\alpha$ and almost every $\gamma\in R$, \begin{equation*} \liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime}=0 \end{equation*} and that there exists $C>0$, such that for all $\alpha\in R\backslash Q$ and $\gamma\in [0,1)$ , \begin{equation*} \liminf_{n\to \infty}n||\gamma -n\alpha||^{\prime} < C. \end{equation*} We prove the first conjecture and disprove the second one