On the Schrödinger Operators with Eigenvalues Embedded into Absolutely Continuous Spectrum

In this paper, we consider the Schrödinger operator on $$L^2(0,\infty )$$ 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 $$L^2(0,\infty )$$ 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.