Imbedded Singular Continuous Spectrum for Schr\"odinger Operators

We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq \frac{h(x)}{1+x},$wherethefunction$h(x)$ is growing arbitrarily slowly, such that the corresponding Schr"odinger operator has imbedded singular continuous spectrum. This solves one of the fifteen “twenty-first century” problems for Schr"odinger operators posed by Barry Simon. The construction also provides the first example of a Schr"odinger operator for which M"oller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.