Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials

Author: Alexei Rybkin
Journal: Proc. Amer. Math. Soc. 130 (2002), 59-67
MSC (2000): Primary 34E05, 34L25; Secondary 34L40
Published electronically: May 10, 2001
MathSciNet review: 1855620
Full-text PDF

Abstract | References | Similar Articles | Additional Information


For the general one dimensional Schrödinger operator $-\frac{d^{2}}{dx^{2}}+q\left(x\right)$ with real $q\in L_{1} \left(\mathbb{R}\right) $ we present a complete streamlined treatment of large spectral parameter power asymptotics of Jost solutions and the scattering matrix. We find simple necessary and sufficient conditions relating the number of exact terms in the asymptotics with the smoothness of $q$. These conditions are expressed in terms of the Fourier transform of some functions related to $q$. In particular, under the usual conditions $q^{\left( N\right) }\in L_{1}\left(\mathbb{R}\right), N\in\mathbb{N} _{0},$ we derive up to two extra terms in the asymptotic expansion of the Jost solution and for the transmission coefficient we derive twice as many terms. Our main results are complete.

References [Enhancements On Off] (What's this?)

  • 1. 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
  • 2. S. Clark and F. Gesztesy, ``Weyl-Titchmarsh $M$-function asymptotics for matrix-valued Schrödinger operators", Proc. London Math. Soc., CMP 20001:02
  • 3. S. Clark, F. Gesztesy, H. Holden, and B. Levitan, ``Borg-type theorems for matrix-valued Schrödinger operators", J. Diff. Eq. 167 (2000), 181-210. CMP 2001:02
  • 4. L.D. Faddeev and V.E. Zakharov, ``Korteweg-De Vries equation: a completely integrable Hamiltonian system", Funct. Anal. Appl. 5, 4 (1971), 280-287.
  • 5. 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
  • 6. 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
  • 7. 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
  • 8. 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. CMP 2001:06
  • 9. F. Gesztesy and B. Simon, ``On local Borg-Marchenko uniqueness results", Comm. Math. Phys. 211 (2000), 273-287. MR 2001b:34020
  • 10. D.B. Hinton, M. Klaus and J.K. Shaw, ``High-energy asymptotics of the scattering matrix on the line", Inverse Problems 5 (1989), 1049-1056. MR 91f:34107
  • 11. D.B. Hinton, M. Klaus and J.K. Shaw, ``Series representation and asymptotics for of the Titchmarsh-Weyl $m$-functions", Differential Integral Equations 2, 4 (1989), 420-429. MR 90m:34057
  • 12. T. Miyazawa, ``Boson representations of one-dimensional scattering", J. Phys. A: Math. Gen. 33 (2000), 191-225. CMP 2000:10
  • 13. A.V. Rybkin, ``On the trace approach to the inverse scattering in dimension one", SIAM J. Math. Anal. 32 (2001), no 6, 1248-1264.
  • 14. A.V. Rybkin, ``A Weyl $m$-function approach to trace formulas for varies Schrödinger operators", in preparation.
  • 15. A.V. Rybkin, ``Some new and old asymptotic representations of the Jost solution and Weyl $m$-function for Schrödinger operators on the line", to appear in Bulletin of LMS.
  • 16. B. Simon, ``A new approach to inverse spectral theory, I. Fundamental formalism", Annals of Math. 150 (1999), 1029-1057. CMP 2000:08
  • 17. E.C. Titchmarsh, ``On eigenfunction expansions associated with second-order differential equations", Oxford University Press, 1950.
  • 18. E.C. Titchmarsh, ``Introduction to the theory of Fourier integrals", Chelsea, New York, 1986. MR 89c:42002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34E05, 34L25, 34L40

Retrieve articles in all journals with MSC (2000): 34E05, 34L25, 34L40

Additional Information

Alexei Rybkin
Affiliation: Department of Mathematical Sciences, University of Alaska–Fairbanks, P.O. Box 756660, Fairbanks, Alaska 99775

Keywords: Schr\"{o}dinger operator, asymptotic expansions, Jost solution, scattering matrix.
Received by editor(s): April 18, 2000
Received by editor(s) in revised form: May 15, 2000
Published electronically: May 10, 2001
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2001 American Mathematical Society