  Introducing a black hole (BH) effective temperature, which takes into account
both the non-strictly thermal character of Hawking radiation and the countable
behavior of emissions of subsequent Hawking quanta, we recently re-analysed BH
quasi-normal modes (QNMs) and interpreted them naturally in terms of quantum
levels. In this work we improve such an analysis removing some approximations
that have been implicitly used in our previous works and obtaining the
corrected expressions for the formulas of the horizon's area quantization and
the number of quanta of area and hence also for Bekenstein-Hawking entropy, its
sub-leading corrections and the number of micro-states, i.e. quantities which
are fundamental to realize the underlying quantum gravity theory, like
functions of the QNMs quantum "overtone" number n and, in turn, of the BH
quantum excited level. An approximation concerning the maximum value of n is
also corrected. On the other hand, our previous results were strictly corrected
only for scalar and gravitational perturbations. Here we show that the
discussion holds also for vector perturbations. The analysis is totally
consistent with the general conviction that BHs result in highly excited states
representing both the "hydrogen atom" and the "quasi-thermal emission" in
quantum gravity. Our BH model is somewhat similar to the semi-classical Bohr's
model of the structure of a hydrogen atom. The thermal approximation of
previous results in the literature is consistent with the results in this
paper. In principle, such results could also have important implications for
the BH information paradox.
