Criteria for embedded eigenvalues for discrete Schrdinger 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), $sigma_{
m ess}(H_0)=sigma_{
m ac}(H)=[-2,2]$ 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 $limsup_{n o infty} n|V(n)|=aotin { E_j}_{j=1}^N+{ E_j}_{j=1}^N$, 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 $0
otin { E_j}+{ E_j}$, 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}

Criteria for embedded eigenvalues for discrete Schrdinger operators Institutional Repository Document