Two- and three-body quantum scattering: Completeness revisited. (English) Zbl 0699.35215

Partial differential equations, Proc. Symp., Holzhau/GDR 1988, Teubner- Texte Math. 112, 108-120 (1989).
[For the entire collection see Zbl 0665.00006.]
The author presents a new simpler and general proof of asymptotic completeness for two and three particle systems. In the “two body” case the Schrödinger operator \[ H=H_ 0+V=(2m)^{-1} \Delta_ x+V(x) \] is considered on the state space \({\mathcal H}=L^ 2(R^{\nu})\), where \(x\in R^{\nu}\) is the relative position of the particles, \(\Delta_ x\) is the Laplacian with respect to x, V(x) is bounded and \[ \sup_{| x| \geq R}| V(x)| =:h(R)\in L^ 1([0,\infty],dR). \] It is shown the existence of \(\lim_{t\to \infty}\exp (iH_ 0t)\exp (- iHt)\psi\) for any \(\psi\) in the continuum spectral subspace. Any state orthogonal to the subspace \({\mathcal H}^{pp}(H)\) of bound states asymptotically moves freely. For the three-particle system it is considered the Schrödinger operator \(H=H_ 0+\sum_{\beta}V^{\beta}(x^{\beta})\) acting on \({\mathcal H}=L^ 2(R^{2\nu})\). Here \(x^{\beta}\in R^{\nu}\) is the relative coordinate for the pair \(\beta\), \(H_ 0=(2\mu^{\beta})^{-1} \Delta_{x^{\beta}}-(2\nu^{\beta})^{-1} \Delta_{y^{\beta \quad}}.\) \(y^{\beta}\) is the position of the third particle relative to the center of mass of the pair and each \(V^{\beta}\) satisfy the same restrictions as in the “two body case”. It is shown that any \(\psi\in {\mathcal H}^{cont}(H)\) can be approximated by \(\psi^ 0+\sum_{\alpha}\psi^{\alpha}\) such that the following limit exists: \[ \lim_{t\to \infty}\exp (iH_ 0t)\exp (-iHt)\psi^ 0,\quad \lim_{t\to \infty}\exp (iH_ 0t)\exp (-iHt)\psi^{\alpha}\quad \forall \alpha. \] Here \(H_{\alpha}:=H_ 0+V^{\alpha}\). Physically this means that all three particles asymptotically move freely relative to each other or a bounded pair moves freely relative to the third particle. The method of proof differs from Sigal’s and Soffer’s approach in the phase-space decomposition which here depends on position velocity and time.
Reviewer: A.B.Borisov


35Q99 Partial differential equations of mathematical physics and other areas of application
81U10 \(n\)-body potential quantum scattering theory
81U05 \(2\)-body potential quantum scattering theory
35J10 Schrödinger operator, Schrödinger equation


Zbl 0665.00006