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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Imbedded singular continuous spectrum for Schrödinger operators
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by Alexander Kiselev PDF
J. Amer. Math. Soc. 18 (2005), 571-603 Request permission

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
  • Email: kiselev@math.wisc.edu
  • 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
  • © Copyright 2005 American Mathematical Society
  • 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
  • MathSciNet review: 2138138