Spinor inverse problems for a gravitational field

Spherically symmetric inverse problems for the scattering of spin-1/2 fermions by a static gravitational field are considered within the framework of the general theory of relativity. Methods are developed for the reconstruction of the metric tensor from the scattering data for the Dirac equation in the Schwarzschild metric. The problem of finding the link between the S-matrix and the Hamiltonian operator in curved space for a neutrino with fixed zero orbital angular momentum and for a massless or massive field with fixed energy is discussed. The main links in the neutrino algorithms consist of two definite systems of two nonlinear ordinary differential equations constructed from the scattering data. It is established, first, that the inverse problems studied in this paper generalize the previously solved classical inverse problems for a gravitational field to the quantum case. Second, they generalize the familiar Marchenko and Regge-Newton methods in quantum theory of scattering to the case of a gravitational field. Third, these inverse problems are a logical extension of the inverse problems for the Klein-Gordon-Fock equation studied in a previous paper. The results may be of interest in astrophysics in connection with the direct determination of the inner structure of objects