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Journal of the American Mathematical Society

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Imbedded singular continuous spectrum for Schrödinger operators

Author: Alexander Kiselev
Journal: J. Amer. Math. Soc. 18 (2005), 571-603
MSC (2000): Primary 34L40; Secondary 34L25
Published electronically: April 27, 2005
MathSciNet review: 2138138
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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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Additional Information

Alexander Kiselev
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388

Keywords: Schrödinger operators, scattering, singular spectrum
Received by editor(s): November 14, 2003
Published electronically: April 27, 2005
Additional Notes: The author was supported in part by NSF grant DMS-0314129 and by an Alfred P. Sloan Research Fellowship
Article copyright: © Copyright 2005 American Mathematical Society