Exact Solutions for the Scalar Field in the Gravitational Field of Spherically Symmetric Black Holes

The problem of finding solutions to the Klein–Gordon equation for a free massive real scalar field in the spacetime of spherically symmetric black holes is considered. It is shown that the radial part of this equation can be reduced to the form of a confluent Heun equation. Physically adequate solutions to this equation expressed through confluent Heun functions are found and their normalizations are calculated. A doubling, compared to the case of the Minkowski space, of the number of physically adequate solutions with energy greater than that of the particle mass has been observed.