Imbedded Singular Continuous Spectrum for Schrödinger Operators

A. Kiselev

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Imbedded Singular Continuous Spectrum for Schrödinger Operators

A. Kiselev

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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 Schrödinger operator has 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 singular continuous spectrum.

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BibTeX RIS

A. Kiselev. 2001-11-19. Imbedded Singular Continuous Spectrum for Schrödinger Operators. https://arxiv.org/abs/math/0111200

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