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Completeness of three body quantum scattering. (English) Zbl 0531.47009

Dynamics and processes, Proc. 3rd Encounter Math. Phys., Bielefeld/Ger. 1981, Lect. Notes Math. 1031, 62-88 (1983).
[For the entire collection see Zbl 0515.00016.]
Let H be three particles quantum Hamiltonian, \(H=\sum_{i}\frac{1}{2m_ i}p^ 2_ i+\sum_{\beta}V_{\beta}(x_{\beta}),\) \(\beta =\{i,j\}\), \(i<j\) where \(V_{\beta}(x)=O(| x|^{-1-\epsilon})\), \(x\cdot\nabla V_{\beta}(x) = o(1)\) as \(| x| \to \infty\), \(\epsilon>0\), \(x_{\beta} = x_ i - x_ j\). Then \(H\) has no singular continuous spectrum and there is the asymptotical completeness property, i.e. \[ \text{Ran}\Omega^0_{\pm} \oplus \oplus_\beta \text{Ran}\Omega^\beta_{\pm} = {\mathcal H}^{ac}(H) \] where \(\Omega^0_{\pm}\), \(\Omega^\beta_{\pm}\) are wave operators corresponding to the separating of all particles and coupling of particles \(i,j\in\beta\) and the separating of the third particle.
Reviewer: V.Ya.Ivrii

MSC:

47A40 Scattering theory of linear operators
81U99 Quantum scattering theory
47L90 Applications of operator algebras to the sciences

Citations:

Zbl 0515.00016