01-425 A. Kiselev

Imbedded Singular Continuous Spectrum for Schr\"odinger Operators (406K, Postscript) Nov 18, 01

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Abstract. We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq \frac{h(x)}{1+x},$ 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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