  In 1976 S. Hawking claimed that "Because part of the information about the
state of the system is lost down the hole, the final situation is represented
by a density matrix rather than a pure quantum state" (Verbatim from ref. 2).
This was the starting point of the popular "black hole (BH) information
paradox". In a series of papers, together with collaborators, we naturally
interpreted BH quasi-normal modes (QNMs) in terms of quantum levels discussing
a model of excited BH somewhat similar to the historical semi-classical Bohr
model of the structure of a hydrogen atom. Here we explicitly write down, for
the same model, a time dependent Schr\"odinger equation for the system composed
by Hawking radiation and BH QNMs. The physical state and the correspondent wave
function are written in terms of an unitary evolution matrix instead of a
density matrix. Thus, the final state results to be a pure quantum state
instead of a mixed one. Hence, Hawking's claim is falsified because BHs result
to be well defined quantum mechanical systems, having ordered, discrete quantum
spectra, which respect 't Hooft's assumption that Schr\"oedinger equations can
be used universally for all dynamics in the universe. As a consequence,
information comes out in BH evaporation in terms of pure states in an unitary
time dependent evolution. In Section 4 of this paper we show that the present
approach permits also to solve the entanglement problem connected with the
information paradox.
