Properties of radial equation of scalar field in Schwarzschild space-time

The scalar field equation is considered in Schwarzschild space-time. The separated radial equation is studied both under the form of Heun differential equation and in the asymptotic behaviour of its solutions. The radial equation is reduced to different canonical forms of confluent Heun differential equation. It is shown that, by series integration and on account of the special dependence of the theory on the physical parameters, Heun's equation does not admit polynomial-like solutions. Moreover, the associated Heun differential operator comes out to be not factorizable. As to the asymptotic behaviour of the solutions, approximated solutions are discussed that lead to a hydrogen like energy spectrum of the system. The corresponding bounded states as well as the general solution are discussed under the action of two scalar products, one of mathematical origin and the other one with the correct covariance property suitable for a quantization of the theory.