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Inverse scattering for N-body systems with time-dependent potentials

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Inverse Problems of Wave Propagation and Diffraction

Part of the book series: Lecture Notes in Physics ((LNP,volume 486))

Abstract

I prove that the high velocity limit of any one of the Dollard scattering operators of an N-body quantum mechanical system with long-range time-dependent pair potentials determines uniquely the potentials. I also show that in the particular case when the potentials go to zero fast enough as time goes to plus and minus infinity it is not necessary to introduce a modified Dollard time evolution and that pair potentials that decrease slowly as the interparticle distances go to infinity (for example Coulomb potentials) can be uniquely reconstructed from the high velocity limit of the canonical scattering operator with unperturbed evolution given by the free Hamiltonian. These results are obtained from reconstruction formulae with bound of the error term that I prove with a simple time-dependent method.

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References

  • Chadan K and Sabatier P. C. (1989): Inverse Problems in Quantum Scattering Theory (2nd edition Springer, Berlin)

    MATH  Google Scholar 

  • Cycon H.L., Froese R. G., Kirsch W. and Simon B. (1987): Schrodinger Operators (Springer, Berlin)

    Google Scholar 

  • Enss V. and Weder R. (1993): Inverse potential scattering:A geometrical approach,in: Mathematical Quantum Theory II:Schrödinger Operators, Proceedings of the Summer School in Mathematical Quantum Theory, August 1993, Vancouver, B. C., Feldman J., Froese R. G. and Rosen L., editors (CRM Proceedings and Lecture Notes 8 AMS, Providence (1995)), 151–162

    Google Scholar 

  • Enss V. and Weder R. (1994): Uniqueness and reconstruction formulae for inverse N-particle scattering, in: Differential Equations and Mathematical Physics, Proceedings of the International Conference, University of Alabama at Birmingham, March 1994, Knowles I., editor (International Press, Boston (1995)), 55–66

    Google Scholar 

  • Enss V. and Weder R. (1995a): The geometrical approach to multidimensional inverse scattering. J. Math. Phys. 36, 3902–3921

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Enss V. and Weder R. (1995b): Inverse two-cluster scattering. Inverse Problems 12, 409–418

    Article  ADS  MathSciNet  Google Scholar 

  • Graf G.M. (1990): Phase space analysis of the charge transfer model. Helv. Phys. Acta 63, 107–138

    MathSciNet  MATH  Google Scholar 

  • Helgason S. (1984): Groups and Geometric Analysis (Academic Press, Orlando)

    MATH  Google Scholar 

  • Ito H. T. (1993): Charge transfer model and (two-cluster) → (two-cluster) three-body scattering. J. Math. Kyoto Univ. 33, 65–113

    MATH  MathSciNet  Google Scholar 

  • Ito H. T. (1995): Charge transfer model and three-body scattering. J. Math. Phys. 36, 115–132

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Kato T. (1953): Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan 5, 200–234

    Article  MathSciNet  Google Scholar 

  • Kato T. (1970): Linear evolution equations of hyperbolic type I. J. Fact. Sci. Univ. of Tokyo Sect IA Math. 17, 241–258

    MATH  Google Scholar 

  • Kato T (1976): Perturbation Theory for linear Operators (2nd edition, Springer, Berlin)

    MATH  Google Scholar 

  • Kitada H. and Yajima K. (1982): A scattering theory for time-dependent long-range potentials. Duke Math. J. 49, 341–376

    Article  MATH  MathSciNet  Google Scholar 

  • Kitada H. and Yajima K. (1983): Remarks on our paper A scattering theory for time-dependent long-range potentials. Duke Math. J. 50, 1005–1016

    Article  MATH  MathSciNet  Google Scholar 

  • Newton R. G. (1989): Inverse Schrödinger Scattering in three Dimensions (Springer, Berlin)

    MATH  Google Scholar 

  • Perla Menzala G. (1985): Sur l'opérateur de diffusion pour l'équation des ondes avec des potentiels dépendant du temps. C.R. Acad. Sc. Paris 300, 621–624

    MATH  MathSciNet  Google Scholar 

  • Ramm A. G. and Sjöstrand J. (1991): An inverse problem for the wave equation. Math. Z. 206, 119–130

    Article  MATH  MathSciNet  Google Scholar 

  • Reed M. and Simon B. (1975): Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness (Academic Press, New York)

    MATH  Google Scholar 

  • Reed M. and Simon B. (1979): Methods of Modern Mathematical Physics III:Scattering Theory (Academic Press, New York)

    MATH  Google Scholar 

  • Stefanov P. D. (1989): Uniqueness for the multi-dimensional inverse scattering problem for time-dependent potentials. Math. Z. 201, 541–559

    Article  MATH  MathSciNet  Google Scholar 

  • Weder R. (1995): Multidimensional inverse scattering in an electric field. J. Func. Anal. 139, 441–465

    Article  MathSciNet  Google Scholar 

  • Wüller U. (1991): Geometric methods in scattering theory of the charge transfer model. Duke Math. J. 62, 273–313

    Article  MATH  MathSciNet  Google Scholar 

  • Yafaev D. R. (1980): Asymptotic completeness for the multidimensional time dependent Schrödinger equation. Sov. Math. Dokl. 21, 545–549

    MATH  Google Scholar 

  • Yajima K. (1980): A multi-channel scattering theory for some time-dependent Hamiltonians, charge transfer model. Comm. Math. Phys. 75, 153–178

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Yajima K. (1987): Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 110, 415–426

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Yajima K. (1991): Schrödinger evolution equations with magnetic fields. J. d'Analyse Math. 56, 29–76

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-726, DF 01000, México

    Ricardo Weder

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  1. Ricardo Weder
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Guy Chavent Pierre C. Sabatier

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© 1997 Springer-Verlag

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Weder, R. (1997). Inverse scattering for N-body systems with time-dependent potentials. In: Chavent, G., Sabatier, P.C. (eds) Inverse Problems of Wave Propagation and Diffraction. Lecture Notes in Physics, vol 486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105758

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  • DOI: https://doi.org/10.1007/BFb0105758

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62865-1

  • Online ISBN: 978-3-540-68713-9

  • eBook Packages: Springer Book Archive

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