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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
DOI: https://doi.org/10.1090/S0894-0347-05-00489-3
Published electronically: April 27, 2005
MathSciNet review: 2138138
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Abstract: We construct examples of potentials $V(x)$ satisfying $\vert V(x)\vert \leq \frac{h(x)}{1+x},$ 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 $\vert V(x)\vert \leq \frac{B}{1+x},$ 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
Email: kiselev@math.wisc.edu

DOI: https://doi.org/10.1090/S0894-0347-05-00489-3
Keywords: Schr\"odinger 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