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 $(H_{0}u)(n)=u(n+1)+u(n-1)$. For $H_0$ (no perturbation), ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(H_0)={\sigma }_{\mathrm{a}\mathrm{c}}(H)=[-2,2]$ 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 ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}n\mathrm{\mid }V(n)\mathrm{\mid }=a<\mathrm{\infty }\mathrm{.}$ 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 $0\mathrm{\notin }\{E_j{\}}_{j=1}^N+\{E_j{\}}_{j=1}^N$, we construct potential $V(n)=\frac{O(1)}{1+\mathrm{\mid }n\mathrm{\mid }}$ 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 $0\mathrm{\notin }\{E_j\}+\{E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct potential $\mathrm{\mid }V(n)\mathrm{\mid }\le \frac{h(n)}{1+\mathrm{\mid }n\mathrm{\mid }}$ such that $H=H_0+V$ has eigenvalues $\{E_j\}$. \end{description}