On the Schrödinger Operators with Eigenvalues Embedded into Absolutely Continuous Spectrum

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.