Scattering matrices in many-body scattering
Let \(\{X_a; a\in I\}\) be a family of linear subspaces of \(\mathbb{R}^n= X_0\), \(X_1= \{0\}\), and \(X^a\) the orthocomplement of \(X_a\). The author treats \(S\)-matrices of the many-body Hamiltonian \(H= \Delta+\sum_{a\in I}(\pi^a)^* V_a\). Consider \(h_a= \Delta+ \sum_{b\leq a}V_b\) on \(\mathbb{L}^2(X^a)\). Let \(\varepsilon_\alpha\), \(\alpha= (a,m)\), be the eigenvalue of \(h_a\) with multiplicity \(m\), \(\psi_\alpha\) the normalized eigenfunction of \(h_a\), and \(H_a= \Delta_{X_a}\otimes \text{Id}+\text{Id}\otimes h_a\). Then \(W^{\pm}_\alpha= s-\lim_{t\to\pm\infty} e^{iHt} e^{-iH_at}J_a\), by \(J_af= ((\pi^a)^* \psi_\alpha)(\pi^*_af)\), and \(S_{\beta\alpha}= (W^+_\beta)^*(W^-_\alpha)\). Standard \(S\)-matrices are \(\widehat S_{\beta\alpha}(\lambda)= {\mathcal F}_\beta S_{\beta\alpha}{\mathcal F}^*_\alpha\) by unitary Fourier transform \({\mathcal F}_\alpha\). The other definition of \(S\)-matrices is given as follows: Using the form of the unique generalized eigenfunction \(u\) of \(H\) relating to \((\alpha,g)\), the author gets \(Q^0_{\beta,\pm}(\pi_\beta u,h)\) from the distributional asymptotic behavior of the projection \(\pi_\beta u\). Then the \(S\)-matrices are \(S_{\beta\alpha}(\lambda)(g)(h)= Q^0_{\beta,+}(\pi_\beta u,h)\). Main Theorem: If \(\alpha\) and \(\beta\) are non-threshold, \(\widehat S_{\beta\alpha}(\lambda)= cS_{\beta\alpha}(\lambda)R\) for \((\Lambda\not\ni) \lambda> \max(\varepsilon_\alpha, \varepsilon_\beta)\), where \(R\) is the pull back by an antipodal map on \(C_a= \overline X_a\cap \partial\mathbb{S}^n_+\).
scientific article; zbMATH DE number 1271631