Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional Schrödinger operators
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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 $q$.References
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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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 219-229
- MSC (1991): Primary 34L40, 47E05; Secondary 34E40
- DOI: https://doi.org/10.1090/S0002-9939-02-06555-3
- MathSciNet review: 1929041