On the long-range scattering for one- and two-particle Schrödinger operators with constant magnetic fields (Q1916080)

The asymptotic completeness is proven of the scattering system \(\{H,H_0\}\) where \(H_0=\frac{1}{2} (-i\nabla_r- \frac{1}{2} B\times r)^2\) and \(H=H_0+V\). The assumptions are: 1) \(B\) is a constant magnetic field, \(B\in\mathbb{R}^3\), \(B\neq 0\). 2) \(V\) is a real valued smooth long-range potential satisfying (i) \(|V(r)|+|r_\parallel \partial_\parallel V(r)|= o(1)\) as \(|r|\to\infty\), \(r\in\mathbb{R}^3\); (ii) \(|\partial_\perp V(r)|\leq C(1+|r_\perp|^2)^{-\frac{1+\delta}{2}}\); (iii) \(|\partial^\ell_\parallel V(r)|\leq C_\ell(1+|r_\parallel|^2)^{-\frac{\ell+\delta}{2}}\). Here \(r_\parallel= \frac{r\cdot B}{|B|}\), \(r_\perp\) denotes the component of \(r\) perpendicular to \(B\); \(\partial_\perp\), and \(\partial_\parallel\) are the partial derivatives with respect to \(r_\perp\) and \(r_\parallel\), respectively; \(\delta\) is some positive constant, \(\ell\) is a positive integer.