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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional Schrödinger operators


Author: Alexei Rybkin
Journal: Proc. Amer. Math. Soc. 131 (2003), 219-229
MSC (1991): Primary 34L40, 47E05; Secondary 34E40
Published electronically: May 1, 2002
MathSciNet review: 1929041
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Abstract: For the general one dimensional Schrödinger operator $-\frac{d^{2}} {dx^{2}}+q\left(x\right)$ with real $q$ we study some analytic aspects related to order-one trace formulas originally due to Buslaev-Faddeev, Faddeev-Zakharov, and Gesztesy-Holden-Simon-Zhao. We show that the condition $q\in L_{1}\left( \mathbb{R}\right) $ guarantees the existence of the trace formulas of order one only with certain resolvent regularizations of the integrals involved. Our principle results are simple necessary and sufficient conditions on absolute summability of the formulas under consideration. These conditions are expressed in terms of Fourier transforms related to $q$.


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Additional Information

Alexei Rybkin
Affiliation: Department of Mathematical Sciences, University of Alaska, Fairbanks, P.O. Box 756660, Fairbanks, Alaska 99775
Email: ffavr@uaf.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06555-3
PII: S 0002-9939(02)06555-3
Keywords: Schr\"{o}dinger operator, Krein's spectral shift function, trace formula.
Received by editor(s): May 8, 2001
Received by editor(s) in revised form: August 31, 2001
Published electronically: May 1, 2002
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society