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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

Citations:

Zbl 0553.00001