On the perturbed periodic Schrödinger operators with separate resonant embedded eigenvalues

In this paper, we consider Schrödinger operators on L^{2}(0,\infty) given by Hu=(H_{0}+V)u=-u^{\prime\prime}+V_{0}u+Vu, where V_{0} is real, 1 -periodic and V is the perturbation. It is well known that under perturbations V(x)=o(1) as x\to\infty , the essential spectrum of H coincides with the essential spectrum of H_{0} . We introduce a new way to construct oscillatory decaying perturbations with resonant embedded eigenvalues. Given any at most countable set S inside the essential spectrum, we can construct perturbations with S contained in the set of eigenvalues if the resonant eigenvalues in S satisfy some condition. In particular, if S is a finite set (or countable set), we can construct perturbation with V(x)=\frac{O(1)}{1+x} (or |V(x)|\leq\frac{h(x)}{1+x} ) as x\to\infty if the resonant eigenvalues of S appear in the same spectral bands or large separate spectral bands, where h(x) is any given function with \lim_{x\to\infty}h(x)=\infty .