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The Schr\"odinger equation of the Schwarzschild black hole (BH) shows that aBH is composed of a particle, the "electron", interacting with a central field,the "nucleus". Via de Broglie's hypothesis, one interprets the "electron" interms of BH horizon's modes. Quantum gravity effects modify the BHsemi-classical structure at the Schwarzschild scale rather than at the Planckscale. The analogy between this BH Schr\"odinger equation and the Schr\"odingerequation of the s states of the hydrogen atom permits us to solve the sameequation. Therefore, BHs are well defined quantum gravitational systems obeyingSchr\"odinger's theory: the "gravitational hydrogen atoms". By identifying thepotential energy in the BH Schr\"odinger equation as being the gravitationalenergy of a spherically symmetric shell, a different nature of the quantum BHseems to surface. BHs are self-interacting, highly excited, sphericallysymmetric, massive quantum shells generated by matter condensing on theapparent horizon, concretely realizing the membrane paradigm. The quantum BHdescripted as a "gravitational hydrogen atom" is a fictitious mathematicalrepresentation of the real, quantum BH, a quantum massive shell having asradius the oscillating gravitational radius. Nontrivial consequences emergefrom this result: i) BHs have neither horizons nor singularities; ii) there isneither information loss in BH evaporation, nor BH complementarity, norfirewall paradox. These results are consistent with previous ones by Hawking,Vaz, Mitra and others. Finally, the special relativistic corrections to the BHSchr\"odinger equation give the BH Klein-Gordon equation and the correspondingeigenvalues.

Schr\"odinger and Klein-Gordon theories of black holes from the quantization of the Oppenheimer and Snyder gravitational collapse