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Schrödinger Operators pp 427–443Cite as

Many-body scattering problem

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Part of the book series: Lecture Notes in Physics ((LNP,volume 345))

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References

  • S. Agmon 1982 Lectures on exponential decay of second-order elliptic equations, Princeton Math. Notes 29.

    Google Scholar 

  • J. Aguilar and J.-M. Combes 1971 A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22, 269–275.

    Google Scholar 

  • J.E. Avron 1981 Bender-Wu formulas for the Zeeman effect in hydrogen, Ann. Phys. 131, 73–94.

    Google Scholar 

  • J.E. Avron and I.W. Herbst 1977 Spectral and scattering theory of Schrödinger operators related to the Stark effect, Commun. Math. Phys. 52, 239–254.

    Google Scholar 

  • J.E. Avron, I. Herbst and B. Simon 1981 Schrödinger operators with magnetic fields, III, Comm. Math. Phys. 79, 529–572.

    Google Scholar 

  • E. Balslev and J.-M. Combes 1971 Spectral properties of Schrödinger Hamiltonians with dilation analytic potentials, Comm. Math. Phys. 22, 280–294.

    Google Scholar 

  • Ph. Briet, J.-M. Combes and P. Duclos 1987a On the location of resonances for Schrödinger operators in the semi-classical limit I: Resonance free domains, J. Math. Anal. Appl. 126, 90–99.

    Google Scholar 

  • Ph. Briet, J.-M. Combes and P. Duclos 1987b On the location of resonances for Schrödinger operators in the quasiclassical limit II: Top resonances, Comm. Part. Diff. Eqn. 12, 201–222.

    Google Scholar 

  • Ph. Briet, J.-M. Combes and P. Duclos 1988a Spectral stability under tunneling, preprint.

    Google Scholar 

  • Ph. Briet, J.-M. Combes and P. Duclos' 1988b On the location of resonances for Schrödinger operators in the semi-classical limit III: The shape resonances, in preparation.

    Google Scholar 

  • A. Bohm 1986 Quantum Mechanics, Springer-Verlag.

    Google Scholar 

  • J.-M. Combes 1974a Spectral deformation in techniques and applications to N-body Schrödinger operators, in Proc. Int. Congress of Mathematicians, Vancouver, 369–376.

    Google Scholar 

  • J.-M. Combes 1974b Analytic perturbation approach to N-particle quantum systems, in Proc. Nato Inst. on Scattering Theory, ed. J.P. Marchard and J. La Vita.

    Google Scholar 

  • J.-M. Combes 1978 Scattering theory in quantum mechanics and asymptotic completeness, in Math. Problems in Theor. Phys., eds. J.G. Del Antonio, Lecture Notes in Physics 80, 183–204.

    Google Scholar 

  • J.-M. Combes 1980 Recent developments in quantum scattering theory, in Math. Problems in Theor. Phys., ed. K. Osterwalder, Lecture Notes in Physics N116, 1–24.

    Google Scholar 

  • J.-M. Combes, P. Duclos, M. Klein and R. Seiler 1987 The shape resonances, Comm. Math. Phys. 110, 215–236.

    Google Scholar 

  • M. Combescure 1986 Long range scattering for the two-body Schrödinger equation with ‘Hörmander like’ potentials, Comm. PDE 11(2), 123–165.

    Google Scholar 

  • H.L. Cycon 1985 Resonances defined by modified dilations, Helv. Phys. Acta 53, 969–981.

    Google Scholar 

  • H. Cycon, R. Froese, W. Kirsch and B. Simon 1987 Schrödinger Operators, Springer-Verlag.

    Google Scholar 

  • S. De Bièvre and P.D. Hislop 1988 Spectral resonances for the Laplace-Beltrami operator, Ann. Inst. H. Poincare 48, 105–145.

    Google Scholar 

  • J. Derezinski 1988a A new proof of propagation theorem for N-body quantum systems, preprint.

    Google Scholar 

  • J. Derezinski 1988b The Mourre estimate for dispersive N-body Schrödinger operators, preprint.

    Google Scholar 

  • J. Derezinski 1988c Criteria for the Kato smoothness with respect to a dispersive N-body Schrödinger operator, preprint.

    Google Scholar 

  • V. Enss 1978 Asymptotic completeness for quantum mechanical potential scattering I. Short-range potentials, Comm. Math. Phys. 61, 285–291.

    Google Scholar 

  • V. Enss 1979 Asymptotic completeness for quantum mechanical potential scattering II. Singular and long-range potentials, Ann. Phys. 119, 117–132 and addendum 1979 preprint Bielefeld.

    Google Scholar 

  • V. Enss 1987 Quantum scattering theory for two-and three-body systems with potentials of short and long range, Schrödinger Operators, ed. S. Graffi, Springer Lecture Notes in Mathematics.

    Google Scholar 

  • V. Enss 1988 The N-body scattering theory, preprint Frei Univ. Berlin.

    Google Scholar 

  • R. Froese and P.D. Hislop 1989 Spectral analysis of second-order elliptic operators on non-compact manifolds, Duke Math. J., to appear.

    Google Scholar 

  • R. Froese and P.D. Hislop 1988 Analysis of the point spectrum of second order elliptic operators on non-compact manifolds, preprint.

    Google Scholar 

  • R. Froese and I. Herbst 1982a Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators, Comm. Math. Phys. 87, 429–447.

    Google Scholar 

  • R. Froese and I. Herbst 1982b A new proof of the Mourre estimate, Duke Math. J. 49, 1075–1085.

    Google Scholar 

  • C. Gérard and J. Sjöstrand 1987 Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys. 108, 391–421.

    Google Scholar 

  • J. Ginibre 1977 Spectral and scattering theory of the Schrödinger equation for three-body systems, in The Schrödinger Equation, Thirring and Urban (eds.).

    Google Scholar 

  • S. Graffi and V. Grecchi 1981 Comm. Math. Phys. 79, 91–109.

    Google Scholar 

  • G. Hagedorn 1979 A link between scattering resonances and dilation analytic resonances in few body quantum mechanics, Comm. Math. Phys. 65, 181–188.

    Google Scholar 

  • E.M. Harrell II and B. Simon 1980 Duke Math. J. 47, 845–902.

    Google Scholar 

  • B. Helffer and A. Martinez 1987 Comparison entre les diverses notions de resonances, Preprint Orsay.

    Google Scholar 

  • B. Helffer and J. Sjöstrand 1986 The Mathematical Theory of Resonances, Mem. S.M.F. no. 24/25, Suppl. Bull. S.M.F. 114.

    Google Scholar 

  • I.W. Herbst and B. Simon 1981 Dilation analyticity in constant electric field II. N-body problem, Borel summability, Comm. Math. Phys. 80, 181–216.

    Google Scholar 

  • P. Hislop and I.M. Sigal 1987 Shape Resonances in Quantum Mechanics, in “Proceedings of the Intern. Conf. on Diff. Equations and Math. Physics, I. Knowles, ed. 1986; and Quantum Resonances, Memoir AMS, to appear.

    Google Scholar 

  • L. Hörmander 1976 The existence of wave operators in scattering theory, Math. Z. 146, 69–91.

    Google Scholar 

  • L. Hörmander 1985 The Analysis of Linear Partial Differential Operators IV, Springer-Verlag.

    Google Scholar 

  • W. Hunziker 1986 Distortion analyticity and molecular resonance curves, Ann. Inst. H. Poincaré 45, 339–358.

    Google Scholar 

  • W. Hunziker 1988 Notes on asymptotic perturbation theory ..., Helv. Phys. Acta 61, 257–304.

    Google Scholar 

  • H. Isozaki and H. Kitada 1985 Modified wave operators with time independent nolifiers, J. Fac. Sci. Univ. Tokyo, Sect. 1A, 32, 77–104.

    Google Scholar 

  • A. Jensen 1977 Local distortion technique, resonances and poles of the S-matrix, J. Math. Anal. Appl. 59, 505–513.

    Google Scholar 

  • T. Kato 1968 Smooth operators and commutators, Studio Mathematics XXXI, 585–646.

    Google Scholar 

  • H. Kitada 1977 Scattering theory for Schrödinger operators with long-range potentials I: Abstract theory, J. Math. Soc. Japan 29, 665–691.

    Google Scholar 

  • H. Kitada 1978 Scattering theory for Schrödinger operators with long-range potentials II: Spectral and scattering theory, J. Math. Soc. Japan 30, 603–632.

    Google Scholar 

  • H. Kitada Theory for Schrödinger operators with time-dependent potentials of long-range type, J. Fac. Sci. Univ. Tokyo, Sect IA, 29, 353–369.

    Google Scholar 

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

    Google Scholar 

  • H. Kitada and K. Yajima 1983 A scattering theory for time-dependent long-range potentials, Duke Math. J. 50, 1005–1016.

    Google Scholar 

  • L. Landau and E. Lifshitz 1977 Quantum Mechanics, Pergamon Press, N.Y.

    Google Scholar 

  • E. Mourre 1981 Absence of singular spectrum for certain self-adjoint operators, Commun. Math. Phys. 78, 391–408.

    Google Scholar 

  • E. Mourre 1983 Operateurs conjugues et proprietes de propagation, Commun. Math. Phys. 91, 279–300.

    Google Scholar 

  • Pl. Muthuramalingam 1984 Spectral properties of vaguely elliptic pseudodifferential operators with momentum-dependent long-range potentials using time-dependent scattering theory, J. Math. Phys. 25, 1881–1899 and Bielefeld preprint (1984).

    Google Scholar 

  • J. Nuttall 1967 Analytic continuation of the off-energy-shell scattering amplitude, J. Math. Phys. 8, 873–877.

    Google Scholar 

  • R. Oppenheimer 1928 Phys. Rev. 31, 66–81.

    Google Scholar 

  • P.A. Perry 1981 Propagation of states in dilation analytic potentials and asymptotic completeness, Comm. Math. Phys. 81, 243–259.

    Google Scholar 

  • P.A. Perry, I.M. Sigal and B. Simon 1981 Spectral analysis of N-body Schrödinger operators, Ann. of Math. 114, 519–567.

    Google Scholar 

  • M. Reed and B. Simon 1975 Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press.

    Google Scholar 

  • Y. Saito 1977 On the asymptotic behaviour of the solutions of the Schrödinger equation (−Δ + Q(v) − k2)V = F, Osaka J. Math. 14, 11–35.

    Google Scholar 

  • Y. Saito 1977 Eigenfunction expansion for the Schrödinger operators with long-range potentials Q(y) = O(|y|−ɛ), ɛ > 0, Osaka J. Math. 14, 37–53.

    Google Scholar 

  • I.M. Sigal 1983a Geometric parametries and the many-body Birman-Schwinger principle, Duke Math. J. 50, 517–537.

    Google Scholar 

  • I.M. Sigal 1983b Mathematical Aspects of Quantum-Mechanical Scattering for Many-Body Systems, Springer Lecture Notes in Mathematics N1011.

    Google Scholar 

  • I.M. Sigal 1984 Complex transformation method and resonances in one-body quantum systems, Ann. Inst. H. Poincaré 41, 103–114, and Addendum 41, 333.

    Google Scholar 

  • I.M. Sigal 1986 Analytic properties of the scattering matrix ..., Int. Eq. Oper. Theory 9, 134–153.

    Google Scholar 

  • I.M. Sigal 1987a Mathematical questions of quantum many-body theory, Seminaire equations aux derivees partielles 1986–1987, Ecole Polytechnique.

    Google Scholar 

  • I.M. Sigal 1987b Lectures on Scattering Theory, University of Toronto, preprint.

    Google Scholar 

  • I.M. Sigal 1988a Sharp exponential bounds on resonance states and width of resonances, Adv. Appl. Math. 9, 127–166.

    Google Scholar 

  • I.M. Sigal 1988b Geometrical theory of Stark resonances in multielectron systems, Comm. Math. Phys. 119, 287–314.

    Google Scholar 

  • I.M. Sigal 1988c On the long-range scattering, preprint.

    Google Scholar 

  • I.M. Sigal 1989a Stark effect in multielectron systems: non-existence of bound states, Comm. Math. Phys. (to appear).

    Google Scholar 

  • I.M. Sigal 1989b Resonances in many-body systems, in Proceedings of the Mathematical Physics Congress, Swansea, 1988, I. Davis and A. Truman, eds. (to appear).

    Google Scholar 

  • I.M. Sigal and A. Soffer 1986 Asymptotic completeness of short-range many-body systems, Bull. AMS 14, 107–110.

    Google Scholar 

  • I.M. Sigal and A. Soffer 1987 N-particle scattering problem: asymptotic completeness for short-range systems, Annal. Math. 125.

    Google Scholar 

  • I.M. Sigal and A. Soffer 1988 Local decay and velocity estimates, preprint, CALTECH.

    Google Scholar 

  • I.M. Sigal and A. Soffer 1989 Asymptotic clustering of many-body Coulomb-type systems, Invent. Math. (to appear).

    Google Scholar 

  • B. Simon 1972 Quadratic from techniques and the Balslev-Combes theorem, Comm. Math. Phys. 27, 1–9.

    Google Scholar 

  • B. Simon 1973 The theory of resonances for dilation-analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math. (2), 97, 246–274.

    Google Scholar 

  • B. Simon 1978 Resonances and complex scaling: A rigorous overview, Int. J. Quantum Chem. 14, 529–542.

    Google Scholar 

  • B. Simon 1979 The definition of molecular resonance curves by the method of exterior complex scaling, Phys. Letts. 71A, 211–214.

    Google Scholar 

  • B. Simon 1983 Semiclassical analysis of low lying eigenvalues, I, Ann. Inst. H. Poincaré 38, 295–307.

    Google Scholar 

  • B. Simon 1984 Semiclassical analysis of low lying eigenvalues, II: Tunnelling, Ann. Math. 120, 89–118.

    Google Scholar 

  • J. Sjöstrand 1987 Semiclassical resonances generated by non-degenerate critical points, preprint.

    Google Scholar 

  • R.F. Streater 1973 Spin-wave scattering, in Scattering Theory in Mathematical Physics, eds. Lavita and Marchand, Riedel Publ. Co., 53–78.

    Google Scholar 

  • W. Thirring and P. Urban eds. 1977 The Schrödinger Equation, Springer-Verlag, N.Y.

    Google Scholar 

  • L.E. Thomas 1973 On the spectral properties of some one-particle Schrödinger Hamiltonians, Helv. Phys. Acta 45, 1057–1065.

    Google Scholar 

  • E.C. Titchmarsh 1949 Proc. Roy. Soc. A200, 39–46.

    Google Scholar 

  • E.C. Titchmarsh 1950 Proc. Roy. Soc. A201, 473–479.

    Google Scholar 

  • E.C. Titchmarsh 1951 Proc. Roy. Soc. A207, 321–328.

    Google Scholar 

  • E.C. Titchmarsh 1954-56 J. Anal. Math. 4, 187–208.

    Google Scholar 

  • E.C. Titchmarsh 1955 Proc. Lond. Math. Soc. 5, 1–21.

    Google Scholar 

  • E.C. Titchmarsh 1958 Eigenfunction Expansions Associated with Second Order Differential Equations, Oxford University Press.

    Google Scholar 

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  1. Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Canada

    I. M. Sigal

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  1. I. M. Sigal
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Helge Holden Arne Jensen

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© 1989 Springer-Verlag

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Sigal, I.M. (1989). Many-body scattering problem. In: Holden, H., Jensen, A. (eds) Schrödinger Operators. Lecture Notes in Physics, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51783-9_27

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  • DOI: https://doi.org/10.1007/3-540-51783-9_27

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