## Regularization of many-particle scattering.(English)Zbl 0617.47006

Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 1149-1159 (1984).
[For the entire collection see Zbl 0553.00001.]
The author reviews the method of unitary regularization in many-particle scattering developed by himself and others, but here exclusively for cases without discrete spectrum of subsystems. The wave operator is given by $$W=s-\lim_{t\to -\infty}\exp (it H)\exp (-it H_ 0)$$ for an n- particle system in d-dimensional space. The Hamiltonian $$H=H_ 0+V$$ $$(V=\sum v(x_ i-x_ j)$$, $$x_ i\in {\mathbb{R}}^ d$$, $$H_ 0=- \sum^{N}_{i}\Delta_ i)$$ is assumed to be selfadjoint. In the center of inertia frame, W is generated by the kernel $(1)\quad W(p,p')=\delta (p-p')-T(p,p')(p^ 2-p^{'2}-i0)^{-1},\quad p\in {\mathbb{R}}^{(n-1)d}.$ For potential scattering and $$n=2$$, T(p,p’) is a smooth function, but, for $$n>2$$, T becomes a distribution with some characteristic singularities, i.e., ”singularities of many-particle scattering”. The regularization starts from heuristic constructions of scattering theory and the corresponding structure of the singularities derived from the properties of kernels T for the subsystem, and hence we expect the singularities of T for the total system to coincide with those of some known kernel $$T_ a$$ and $$T-T_ a$$ to have the prescribed smoothness. Steps of the construction give a unitary operator $$W_ a$$ connected to $$T_ a$$ by formula (1), where $$T_ a$$ differs from the original kernel only by a smooth definite addend. Introducing $$\tilde V=W^*_ aK$$, $$K=HW_ a-W_ aH_ 0$$, we have $$H_ a=H_ 0+\tilde V$$. Under certain assumptions of smoothness on some kernels constructed from $$T_ a$$ we can assert the existence of $$U=s-\lim_{t\to -\infty}\exp (it H_ a)\exp (-it H_ 0)$$ and the equality $$W=W_ aU$$. These assure the original heuristic conception of the scattering theory under the assumptions referred to above. This review paper also covers articles on one-dimensional particles and trace formulas.

### MSC:

 47A40 Scattering theory of linear operators 81U10 $$n$$-body potential quantum scattering theory

Zbl 0553.00001