Imbedded Singular Continuous Spectrum for Schr\"odinger Operators
Abstract
We construct examples of potentials V(x) satisfying |V(x)| ≤ h(x)1+x, where the function 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.
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