  We analyze the absorption of microwaves by the Heisenberg-Ising chain
combining exact calculations, based on the integrability of the model, with
numerical calculations. Within linear response theory the absorbed intensity is
determined by the imaginary part of the dynamical susceptibility. The moments
of the normalized intensity can be used to define the shift of the resonance
frequency induced by the interactions and the line width independently of the
shape of the spectral line. These moments can be calculated exactly as
functions of temperature and strength of an external magnetic field, as long as
the field is directed along the symmetry axis of the chain. This allows us to
discuss the line width and the resonance shift for a given magnetic field in
the full range of possible anisotropy parameters. For the interpretation of
these data we need a qualitative knowledge of the line shape which we obtain
from fully numerical calculations for finite chains. Exact analytical results
on the line shape are out of reach of current theories. From our numerical work
we could extract, however, an empirical parameter-free model of the line shape
at high temperatures which is extremely accurate over a wide range of
anisotropy parameters and is exact at the free fermion point and at the
isotropic point. Another prediction of the line shape can be made in the
zero-temperature and zero magnetic field limit, where the sufficiently
anisotropic model shows strong absorption. For anisotropy parameters in the
massive phase we derive the exact two-spinon contribution to the spectral line.
From the intensity sum rule it can be estimated that this contribution accounts
for more than 80% of the spectral weight if the anisotropy parameter is
moderately above its value at the isotropic point.
