High-energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and applications

Abstract

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short-range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high-energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy $E.$ Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in $L^{2}\left( \Omega\right) ,$ where $\Omega$ is any connected bounded open set in $\mathbb{R}^{3}$ with smooth boundary, and we show that if we know an electric potential and a magnetic field for $\mathbb{R}^{3}\setminus\Omega$, then the scattering amplitude, given for some energy $E$, uniquely determines these electric potential and magnetic field everywhere in $\mathbb{R}^{3}$. Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some $E$, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field, respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge-conjugation and time-reversal transformations for the Dirac operator.