Journal of the American Mathematical Society

Author(s): Alexander Kiselev
Journal: J. Amer. Math. Soc. 18 (2005), 571-603.
MSC (2000): Primary 34L40; Secondary 34L25
Posted: April 27, 2005
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Abstract: We construct examples of potentials

satisfying $\vert V(x)\vert \leq \frac{h(x)}{1+x},$ where the function

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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Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 34L40, 34L25

Alexander Kiselev
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
Email: kiselev@math.wisc.edu

DOI: 10.1090/S0894-0347-05-00489-3
PII: S 0894-0347(05)00489-3
Keywords: Schr\"odinger operators, scattering, singular spectrum
Received by editor(s): November 14, 2003
Posted: 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 of article: Copyright 2005, American Mathematical Society

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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