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

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