  Using Thorne's hoop conjecture, it is argued that the energies of the Hawking
quanta emitted from canonical Schwarzschild black holes are bounded from above
by the simple quantum relation ${\cal E}<{\cal
E}_{\text{max}}={\hbar^{2/3}}/M^{1/3}$. In particular, it is shown that, due to
non-linear (self-gravity) effects of the tunneling quanta, higher energy field
modes are re-absorbed (rather than escape to infinity) by the black hole.
