## 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.## References

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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