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New channels of scattering for three-body quantum systems with long-range potentials. (English) Zbl 0859.35084

The author shows that, for some three-body systems with sufficiently slowly decreasing pair potentials (decaying for some of them like \(\langle y\rangle^{-\varrho}\) with \(\varrho\in (0,1/2)\) at infinity), there exist new channels of scattering which were not present when \(\varrho>1/2\).
Reviewer: B.Helffer (Orsay)

MSC:

35P25 Scattering theory for PDEs
81U10 \(n\)-body potential quantum scattering theory
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[1] L. Hörmander, The analysis of linear partial differential operators. IV , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. · Zbl 0612.35001
[2] L. D. Faddeev, Mathematical aspects of the three-body problem in quantum scattering theory , Trudy Mat. Inst. Steklov. 69 (1963), (in Russian); English translation by Ch. Gutfreund, Daniel Davey & Co., Inc., New York, 1965. · Zbl 0131.43504
[3] V. Enss, Completeness of three-body quantum scattering , Dynamics and processes (Bielefeld, 1981), Lecture Notes in Math., vol. 1031, Springer, Berlin, 1983, pp. 62-88. · Zbl 0531.47009 · doi:10.1007/BFb0072111
[4] V. Enss, Quantum scattering theory for two- and three-body systems with potentials of short and long range , Schrödinger operators (Como, 1984), Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 39-176. · Zbl 0585.35023
[5] V. Enss, Long-range scattering of two- and three-body quantum systems , Journées “Équations aux Dérivées Partielles” (Saint Jean de Monts, 1989), École Polytech., Palaiseau, 1989, p. 31. · Zbl 0734.35069
[6] X. P. Wang, On the three-body long-range scattering problems , Lett. Math. Phys. 25 (1992), no. 4, 267-276. · Zbl 0776.35045 · doi:10.1007/BF00398399
[7] C. Gérard, Asymptotic completeness for \(3\)-particle long-range systems , Invent. Math. 114 (1993), no. 2, 333-397. · Zbl 0812.70006 · doi:10.1007/BF01232674
[8] J. Dereziński, Asymptotic completeness of long-range \(N\)-body quantum systems , Ann. of Math. (2) 138 (1993), no. 2, 427-476. JSTOR: · Zbl 0844.47005 · doi:10.2307/2946615
[9] D. R. Yafaev, On the break-down of completeness of wave operators in potential scattering , Comm. Math. Phys. 65 (1979), no. 2, 167-179. · Zbl 0401.47006 · doi:10.1007/BF01225147
[10] D. R. Yafaev, New channels in two-body long-range scattering , to appear in Algebra and Analysis 8, 1996. · Zbl 0854.35080
[11] D. R. Jafaev, Wave operators for the Schrödinger equation , Teoret. Mat. Fiz. 45 (1980), no. 2, 224-234, in Russian. · Zbl 0458.35078
[12] J. Dollard, Asymptotic convergence and the Coulomb interaction , J. Mathematical Phys. 5 (1964), 729-738. · doi:10.1063/1.1704171
[13] V. S. Buslaev and V. B. Matveev, Wave operators for the Schrödinger equation with slowly decreasing potential , Teoret. Mat. Fiz. 2 (1970), no. 3, 367-376.
[14] D. R. Yafaev, The low energy scattering for slowly decreasing potentials , Comm. Math. Phys. 85 (1982), no. 2, 177-196. · Zbl 0509.35065 · doi:10.1007/BF01254456
[15] M. S. Ashbaugh and E. M. Harrell, Perturbation theory for shape resonances and large barrier potentials , Comm. Math. Phys. 83 (1982), no. 2, 151-170. · Zbl 0494.34044 · doi:10.1007/BF01976039
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