Quantum scattering theory for two- and three-body systems with potentials of short and long range.

*(English)*Zbl 0585.35023
Schrödinger operators, Lect. 2nd 1984 Sess. C.I.M.E., Como/Italy, Lect. Notes Math. 1159, 39-176 (1985).

[For the entire collection see Zbl 0565.00010.]

This paper is an expanded version of a series of lectures delivered by the author at the CIME session in 1984. It contains a detailed exposition of the results that he obtained in two-body and three-body quantum mechanical scattering theory by the time-dependent method that he initiated in 1978.

The main result is a proof of asymptotic completeness (A.C.), including the absence of singular continuous spectrum, both for two-body and three- body potentials. It covers the case of short range and of long range potentials V, with the free evolution being suitably modified in the latter case. Here short range means that V decreases faster than \(| x|^{-1}\), in a suitable integrable sense and long range means that \(\nabla V\) decreases faster than \(| x|^{-3/2}\) (and therefore V faster than \(| x|^{-1/2})\) in most of the argument. The proof of A.C. in the long range case given here is the main new result of this paper. The method makes an essential use of propagation properties of scattering states in the phase space of the system and requires a large number of estimates giving a quantitative meaning to the intuitive physical description of scattering processes. The paper contains an extensive bibliography.

This paper is an expanded version of a series of lectures delivered by the author at the CIME session in 1984. It contains a detailed exposition of the results that he obtained in two-body and three-body quantum mechanical scattering theory by the time-dependent method that he initiated in 1978.

The main result is a proof of asymptotic completeness (A.C.), including the absence of singular continuous spectrum, both for two-body and three- body potentials. It covers the case of short range and of long range potentials V, with the free evolution being suitably modified in the latter case. Here short range means that V decreases faster than \(| x|^{-1}\), in a suitable integrable sense and long range means that \(\nabla V\) decreases faster than \(| x|^{-3/2}\) (and therefore V faster than \(| x|^{-1/2})\) in most of the argument. The proof of A.C. in the long range case given here is the main new result of this paper. The method makes an essential use of propagation properties of scattering states in the phase space of the system and requires a large number of estimates giving a quantitative meaning to the intuitive physical description of scattering processes. The paper contains an extensive bibliography.

Reviewer: J.Ginibre

##### MSC:

35J10 | Schrödinger operator, Schrödinger equation |

47A40 | Scattering theory of linear operators |

35P25 | Scattering theory for PDEs |

81U05 | \(2\)-body potential quantum scattering theory |