## 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

### Keywords:

three particles quantum Hamiltonian; wave operators

Zbl 0515.00016