Quantum mechanics in Riemann space-time

Asymptotics in c-2 (c being the light velocity) is studied for the theory of a complex scalar field in Riemannian space-time interacting with an external electromagnetic field. A general-covariant analogue of the Schroedinger equation is obtained for a scalar particle in external gravitational and electromagnetic fields with the relativistic corrections of arbitrary orders in the normal Gaussian reference system. It is shown that taking into account the geometrical change in time of the spatial volume element leads to Hamiltonian which is (asymptotically) self-adjoint with respect to the standard scalar product. This fact is considered as an argument in favour of the Born interpretation of the wave functions