Criteria for Embedded Eigenvalues for Discrete Schrodinger Operators

In this paper, we consider discrete Schrodinger operators of the form, (Hu) (n) = u(n + 1) + u(n - 1) + V(n)u(n). 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), sess(H-0) = sac(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 Prufer 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 lim sup(n ->infinity) n|V(n)| = a < infinity. 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 is not an element of {E-j}(j=1)(N) + {E-j}(j=-1)(N), we construct the explicit potential V(n) = 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 is not an element of {E-j} + {E-j}, and any function h(n) > 0 going to infinity arbitrarily slowly, we construct the explicit potential |V(n)| = h(n)/1+|n| such that H = H-0 + V has eigenvalues {E-j}.