A regularization method is presented to deduce dynamic correlation functions from exact diagonalization calculations. It is applied to dimer-dimer correlation functions in quantum spin chains relevant for the description of spin-Peierls systems. Exact results for the XY model are presented. The analysis draws into doubt that the dimer-dimer correlation functions show the same scale invariance as spin-spin correlation functions. The results are applied to describe the quasi-elastic scattering in CuGeO$_3$.