  The Schrodinger equation of the Schwarzschild black hole (SBH) has been
recently found by the Author and collaborators. By following with caution the
analogy between this SBH Schrodinger equation and the traditional Schrodinger
equation of the s states (l=0) of the hydrogen atom, the SBH Schrodinger
equation can be solved and discussed. The approach also permits to find the
quantum gravitational quantities which are the gravitational analogous of the
fine structure constant and of the Rydberg constant. Remarkably, such
quantities are not constants. Instead, they are dynamical quantities having
well defined discrete spectra. In particular the spectrum of the "gravitational
fine structure constant" is exactly the set of non-zero natural numbers
\mathbb{N}-\left\{ 0\right\} . Therefore, one argues the interesting
consequence that the SBH results in a perfect quantum gravitational system,
which obeys Schrodinger's theory: the "gravitational hydrogen atom".
