  The Schr\"odinger equation of the Schwarzschild black hole (BH) shows that a
BH is composed of a particle, the "electron", interacting with a central field,
the "nucleus". Via de Broglie's hypothesis, one interprets the "electron" in
terms of BH horizon's modes. Quantum gravity effects modify the BH
semi-classical structure at the Schwarzschild scale rather than at the Planck
scale. The analogy between this BH Schr\"odinger equation and the Schr\"odinger
equation of the s states of the hydrogen atom permits us to solve the same
equation. Therefore, BHs are well defined quantum gravitational systems obeying
Schr\"odinger's theory: the "gravitational hydrogen atoms". By identifying the
potential energy in the BH Schr\"odinger equation as being the gravitational
energy of a spherically symmetric shell, a different nature of the quantum BH
seems to surface. BHs are self-interacting, highly excited, spherically
symmetric, massive quantum shells generated by matter condensing on the
apparent horizon, concretely realizing the membrane paradigm. The quantum BH
descripted as a "gravitational hydrogen atom" is a fictitious mathematical
representation of the real, quantum BH, a quantum massive shell having as
radius the oscillating gravitational radius. Nontrivial consequences emerge
from this result: i) BHs have neither horizons nor singularities; ii) there is
neither information loss in BH evaporation, nor BH complementarity, nor
firewall paradox. These results are consistent with previous ones by Hawking,
Vaz, Mitra and others. Finally, the special relativistic corrections to the BH
Schr\"odinger equation give the BH Klein-Gordon equation and the corresponding
eigenvalues.
