  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}
