Short-range scattering in long-range magnetic fields: The relativistic case
Consider the Dirac equation in the Hamiltonian form, \(i(d/dt)\psi =h\psi\) for \(\psi\in L^2(R^3)^4\) where \(H=H_0+V\), \(H_0=c\alpha\cdot p+\beta mc^2\), \(p=-i\nabla\). \(\alpha =(\alpha _1,\alpha_2,\alpha_3)\) and \(\beta\) are \(4\times4\) Hamiltonian matrices. c is the velocity of light, \(m\) the mass of the particle. \(V\) is multiplication by a hermitian \(4\times 4\) matrix valued function. The authors prove that the Moller operators \(\Omega_\pm\equiv s- \lim_{t\to+\infty}e^{iHt}e^{- iH_0t}\) exist and are strongly asymptotically complete under the following assumptions: 1. \(| F_{\rho}(x)| \leq const.(1+| x|)^{-3/2- \delta}\), 2. \(\left|\left[L_{\sigma},G_{\rho}\right](x)\right| \leq const.(1+| x|)^{-3/2-\delta}\), 3. \(\left|\left[L_{\sigma},\sum _{\rho}\left[L_{\rho},G_{\rho}\right] \right](x)\right| \leq \text{const.}(1+| x|)^{-1/2- \delta}\) for some \(\delta >0\). \(\sigma\),\(\rho\) denote any pair of ordered indices \((i,k)\), \(1\leq i<k\leq3\). \(L_{\sigma}\) are the angular momenta \(L_{ik}=x_ ip_ k-x_ kp_ i\), where \(G_{ik}(x)=\sum ^{3}_{k=1}\int^{1}_{0}ds\quad sF_{ik}(xs)\) and \(F_{ik},i,k=1,2,3\) is the magnetic field (or Yang-Milles field). 4. The vector potential \(A\) is the ``transversal gauge'' defined by \(A(x).x=0\) and \(V(x)=-c\alpha.A(x)\).
scientific article; zbMATH DE number 4086093