Criteria for Embedded Eigenvalues for Discrete Schrodinger Operators
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AbstractIn this paper, we consider discrete Schrdinger 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), $sigma _{ extrm{ess}}(H_0)=sigma _{ extrm{ac}}(H)=[-2,2]$ 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 almost sign type potentials and develop the Prfer transformation to address this problem, which leads to the following five results.1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $limsup _{n o infty } n|V(n)|=a>infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points ${ E_j}_{j=1}^N$ in $(-2,2)$ with $0 otin { E_j}_{j=1}^N+{ E_j}_{j=1}^N$, we construct the explicit potential $V(n)=frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues ${ E_j}_{j=1}^N$.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 the explicit potential $|V(n)|leq frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues ${ E_j}$.
