Internal structure of relativistic astrophysical objects in the wave approximation

Spherically symmetric inverse problems for the scattering of quantum particles by a static gravitational field are considered within the framework of general relativity theory. Methods are developed for determining the metric tensor on the basis of scattering data for a fixed energy or zero angular momentum for the Klein-Fock-Gordon equation in the Schwarzschild metric. The relation between the S-matrix and squared 4-momentum operator in curved space is investigated. The main elements of the algorithms developed are two definite nonlinear ordinary differential equations of the third and fourth order based on the scattering data. On the one hand, the inverse problems studied extend the classical inverse problems for the gravitational field, solved previously, to the quantum case. On the other hand, they extend the Marchenko and Regge-Newton methods familiar in quantum theory to the case of a gravitational field. An analogy is established between the motion of a scalar particle in a strong gravitational field and the motion in a field with a potential depending on the angular momentum or energy in nonrelativistic quantum mechanics