Imbedded singular continuous spectrum for Schrödinger operators

We construct examples of potentials $V(x)$ satisfying $\vert V(x)\vert \leq \frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schrödinger operator has an imbedded singular continuous spectrum. This solves one of the fifteen ``twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We also prove that if $\vert V(x)\vert \leq \frac{B}{1+x},$ the singular continuous spectrum is empty. Therefore our result is sharp.