Wave operators for multi-channel long-range scattering

  • A. W. Sáenz
Quantum-Mechanical Scattering Theory
Part of the Lecture Notes in Physics book series (LNP, volume 130)


We consider a multichannel scattering system composed of N ⩾ 2 spinless, distinguishable, nonrelativistic particles, each with configuration space ℝ3 and interacting pairwise by local potentials V ij (1 ⩽ i < jN) consisting of long-range and short-range parts. Each V ij can be chosen, roughly, with the same degree of generality as in ALSHOLM [3], when the configuration space in the latter reference is taken as ℝ3. Using techniques similar to those of ALSHOLM [2,3], we have proved for the class of potentials considered that suitable modified wave operators μ ± α . exist and have a generalized intertwining property for each channel α such that the corresponding bound states have a mild decay property of infinity. For the present class of V ij 's, this property is known to be possessed by those bound states corresponding to eigenvalues of the discrete spectrum of the pertinent cluster Hamiltonian, or even to arbitrary nonthreshold eigenvalues if in addition the V ij 's are dilatation analytic. We have also proved that the usual range-orthogonality property of the μ ± α holds under the conditions stated below. The results of this paper can be readily generalized to the case when the single-particle configuration space is ℝv(v ⩾ 1).


Configuration Space Local Potential Wave Operator Modern Mathematical Physic Asymptotic Completeness 
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Notes and References

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    J.D. Dollard: Ph. D. thesis, Princeton University, 1963; J. Math. Phys. 5, 729 (1964); Rocky Mt. J. Math. 1, 5 (1971) [ Erratum: Rocky Mt. J. of Math. 2, 217 (1972)Google Scholar
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    P. Alsholm: “Wave operators for long-range scattering,” Ph. D. thesis, University of California, Berkeley, 1972Google Scholar
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    P. Alsholm: J. Math. Anal. Appl. 59, 550 (1977)Google Scholar
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    L. Hormänder, Math. Zeits. 146, 69 (1976)Google Scholar
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    For a review of investigations on modified wave operators for single-channel scattering by long-range potentials, see, e.g., W.O. Amrein: in Scattering Theory in Mathematical Physics, ed. by J. LaVita and J.-P. Marchand (Reidel, Dordrecht, Holland, 1974). See also [7]Google Scholar
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    A.M. Berthier, P. Collet: Ann. Inst. Henri Poincaré 26, 279 (1977)Google Scholar
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    M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. III (Academic, New York, 1979), Sec. XI. 9 and Notes to Sec. XI. 9Google Scholar
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    A similar assumption has been made by J. H. Hendrickson, J. Math. Phys. 17, 729 (1976) to establish the existence of modified wave operators for multichannel scattering systems with long-range timedependent potentialsGoogle Scholar
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    For rigorous results on the decay of bound-state wave functions of multiparticle systems and a review of the literature, see M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. IV (Academic, New York, 1978), Sec. XIII.11 and Notes to Sec. XIII.11Google Scholar
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    W.W. Zachary: J. Math. Phys. 17, 1056 (1976) [Erratum: J. Math. Phys. 18, 536 (1977)]Google Scholar
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    In this paper, the ℝ3 case is considered for simplicity. The present work is easily generalized to the case of a single-particle configuration space ℝv(v ⩾ 1) under conditions on the long-range parts of the V ij's roughly the same as in [3].Google Scholar
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    W. Hunziker: J. Math. Phys. 6, 6 (1965)Google Scholar
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    V. Buslaev, V.B. Matveev: Teor. Mat. Fiz. 2, 367 (1970) [Theoret. Math. Phys. 2, 266 (1970)] [14] See, e.g., Theorem XIII.39 of [9]Google Scholar
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    See p. 234 of [91Google Scholar
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    See, e.g., Theorem XIIIA1 of [9]Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • A. W. Sáenz
    • 1
  1. 1.Naval Research LaboratoryWashingto

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