Imbedded singular continuous spectrum for Schrödinger operators

Kiselev, Alexander

doi:10.1090/S0894-0347-05-00489-3

Imbedded singular continuous spectrum for Schrödinger operators
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by Alexander Kiselev

J. Amer. Math. Soc. 18 (2005), 571-603

DOI: https://doi.org/10.1090/S0894-0347-05-00489-3

Published electronically: April 27, 2005

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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 an 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 a singular continuous spectrum. We also prove that if $|V(x)| \leq \frac {B}{1+x},$ the singular continuous spectrum is empty. Therefore our result is sharp.

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