In this paper, we consider the Schrödinger operator on L² (0, ) given by (Hu) (x) =-u'' (x) +V (x) u (x) with a self-adjoint boundary condition at 0, where V (x) is the real perturbation. It is well known that under L² (0, ) perturbations the absolutely continuous spectrum of H on the positive semi-axis is preserved. In this paper, by the technique of modified Prüfer transformation and constructive methods, we prove that, with a class of smooth perturbations, H has exactly the given eigenvalues embedded into the absolutely continuous spectrum.
Lyu et al. (Fri,) studied this question.