High-energy behavior of the scattering amplitude for a Dirac operator
Let \(U_{H_0}: L_2(\mathbb{R}^3;\mathbb{C}^4)\to L_2(\mathbb{R}^3;\mathbb{C}^4)\) denote a spectral representation of the unperturbed Dirac operator \(H_0\) in \(\mathbb{R}^3\), which can be constructed by Fourier transform techniques, i.e. we have that \(U_{H_0}\) is unitary and \(U_{H_0}H_0U^*_{H_0}= E\), where \(E\) denotes the multiplication-by-the-argument operator. The integral kernel \(T\) of the Hilbert-Schmidt operator \(S(E)-1\), where \(S(E):=U_{H_0}SU^*_{H_0}\) denotes the scattering matrix, associated with the scattering operator \(S\) corresponding to the perturbed Dirac operator \(H= H_0+V\), gives rise to the definition of the so-called scattering amplitude \[ F(E,\theta,\omega):=- 2\pi i\;c(E^2-m^2c^4)^{-1/2}T(E,\theta,\omega), \] \(c\) denotes the speed of light, \(m\) the rest mass of the particle. In these multiplication operators, \(E\) is interpreted as a parameter describing the energy level. The asymptotic behavior for large \(E\) of the scattering amplitude \(E\mapsto F(E,\cdot,\cdot)\) and of the total scattering cross section \(E\mapsto\int_{S^2}|F(E,\theta,\omega)|^2d\theta\) is studied for a Dirac operator \(H\) with a time-independent \(4\times 4\)-complex-matrix-valued potential \(V\) satisfying suitable asymptotic conditions. In case that \(V\) is produced by a magnetic potential \(A\) and an electric potential \(\phi\), it is found that \(A\) and \(\phi\) can be reconstructed from the information conserved in the high-frequency behavior of the scattering amplitude \(F\). The results rely on estimates of the -- in the sense of the limiting absorption principle -- extended resolvents \((H-(E+i\;0\pm))^{-1}\) for large parameters \(E\in\mathbb{R}\).
asymptotic behavior of the scattering amplitude
scientific article; zbMATH DE number 906886
scientific article