  We elaborate on the fact that quarkonium in hot QCD should not be thought of
as a stationary bound state in a temperature-dependent real potential, but as a
short-lived transient, with an exponentially decaying wave function. The reason
is the existence of an imaginary part in the pertinent static potential,
signalling the ``disappearance'', due to inelastic scatterings with hard
particles in the plasma, of the off-shell gluons that bind the quarks together.
By solving the corresponding Schr\"odinger equation, we estimate numerically
the near-threshold spectral functions in scalar, pseudoscalar, vector and axial
vector channels, as a function of the temperature and of the heavy quark mass.
In particular, we point out a subtlety in the determination of the scalar
channel spectral function and, resolving it to the best of our understanding,
suggest that at least in the bottomonium case, a resonance peak can be observed
also in the scalar channel, even though it is strongly suppressed with respect
to the peak in the vector channel. Finally, we plot the physical dilepton
production rate, stressing that despite the eventual disappearance of the
resonance peak from the corresponding spectral function, the quarkonium
contribution to the dilepton rate becomes more pronounced with increasing
temperature, because of the yield from free heavy quarks.
