Imbedded Singular Continuous Spectrum for Schrödinger Operators

Details Imbedded Singular Continuous Spectrum for Schrödinger Operators Imbedded Singular Continuous Spectrum for Schrödinger Operators

2001-11-19

arxiv.org/abs/math/0111200v1

Mathematics

Spectral Theory

30 pages, 2 figures

Scientific paper

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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