  We investigate the influence of quenched disorder on the dynamic structure
factor of Heisenberg bilayers on the square, triangular, and kagome lattice in
the quantum paramagnetic phase. Perturbative continuous unitary transformations
and white graphs are employed to calculate the one-triplon contribution up to
high orders in perturbation about the dimer limit for bimodal and continuous
disorder. For the square lattice we find that the lifetime of the gap mode is
increased by stronger quantum correlations while stronger disorder effects are
observed for the triangular lattice due to geometric frustration. For
intra-dimer disorder, in-band energy gaps are observed for both lattices which
can be understood in terms of a level repulsion on dimers with low and high
intra-dimer exchange that are close in energy at the momentum where the in-band
gap opens. For the highly frustrated kagome lattice disorder even allows to
decrease the gap energy. In addition, the localization length of the low-energy
flat band is increased up to order $7$ in perturbation theory. The interplay of
quenched disorder, geometric frustration, and strong correlations leads
therefore to rich structures in the dynamical structure factor of
two-dimensional quantum magnets.
