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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
DOI: https://doi.org/10.1090/S0002-9939-02-06555-3
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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  • 1. F. Atkinson, ``On the location of the Weyl circles'', Proc. Roy. Soc. Edinburgh 88A (1981), 345-356.
  • 2. M.Sh. Birman and D.R. Yafaev, ``The spectral shift function. The work of M.G. Krein and its further development'', St. Petersburg Math. J. 4 (1993), 833-870. MR 94g:47002
  • 3. V.S. Buslaev and L.D. Faddeev, ``Formulas for traces for a singular Sturm-Liouville differential operator'', Soviet Math. Dokl. 1 (1960), 451-454. MR 22:11171
  • 4. L.A. Dikii, ``Trace formulas for Sturm-Liouville differential operators'', Am. Math. Soc. Trans. 18 (1961), 81-115. MR 23:A1874
  • 5. L.D. Faddeev and V.E. Zakharov, ``The Korteweg-De Vries equation is a fully integrable Hamiltonian system'', Funct. Anal. Appl., 5 (1971), 18-27. MR 46:2270
  • 6. I.M. Gel'fand and B.M. Levitan, ``On a simple identity for eigenvalues of a second order differential operator'', Dokl. Akad. Nauk SSSR 88 (1953), 593-596 (in Russian); English translation in I.M. Gel'fand, Collected Papers (Springer Verlag, Berlin, 1987), Vol. 1, 457-461. MR 15:33a
  • 7. F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, ``Trace formulae and inverse spectral theory for Schrödinger operators'', Bull. Am. Math. Soc. 29 (1993), 250-255. MR 94c:34127
  • 8. F. Gesztesy and H. Holden, ``Trace formulas and conservation laws for nonlinear evolution equation'', Rev. Math. Phys. 6, 1 (1994), 51-95. MR 95h:35198a; errata MR 95h:35198b
  • 9. F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, ``Higher order trace relations for Schrödinger operators'', Rev. Math. Phys. 7, 6 (1995), 893-922. MR 97d:34094
  • 10. F. Gesztesy, H. Holden, and B. Simon, ``Absolute summability of the trace relation for certain Schrödinger operators'', Commun. Math. Phys. 168 (1995), 137-161. MR 96b:34110
  • 11. F. Gesztesy and B. Simon,``The $\xi$ function'', Acta Math. 176 (1996), 49-71. MR 97e:47078
  • 12. F. Gesztesy and B. Simon, ``A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure'', Ann. Math. 152 (2000), 593-643. MR 2001m:34185b
  • 13. V.A. Javrjan, ``The spectral shift function for Sturm-Liouville differential operators'', Akad. Nauk Armjan. SSR Dokl 38 (1964), 193-198. MR 29:3710
  • 14. A.V. Rybkin, ``On $A$-integrability of the spectral shift function of unitary operators arising in the Lax-Phillips scattering theory'', Duke Math. J. 83 (1996), 683-699. MR 97e:58224
  • 15. A.V. Rybkin ``KdV invariants and Herglotz functions'', Differential Integral Equations, 14, no. 4 (2001), 493-512. MR 2001k:34155
  • 16. A.V. Rybkin, ``On the trace approach to the inverse scattering in dimension one'', SIAM J. Math. Anal. 32 (2001), 6, 1248-1264.
  • 17. A.V. Rybkin, ``Some new and old asymptotic representations of the Jost solution and Weyl $m$-function for Schrödinger operators on the line'', Bulletin of London Math Soc. 34 (2002), 61-72. CMP 2002:04
  • 18. A.V. Rybkin, ``A Weyl $m$-function approach to trace formulas for various Schrödinger operators'', in preparation.
  • 19. E.C. Titchmarsh, ``On eigenfunction expansions associated with second-order differential equations'' Oxford University Press 1946. MR 8:458d
  • 20. E.C. Titchmarsh, ``Introduction to the theory of Fourier integrals'', Chelsea, New York, 1986. MR 89c:42002
  • 21. T. Weidl, ``On the Lieb-Thirring constants $L_{\gamma,1}$ for $\gamma\geq1/2$'', Commun. Math. Phys. 178 (1996), 135-146. MR 97c:81039

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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: https://doi.org/10.1090/S0002-9939-02-06555-3
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

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