  We study the matrix elements of local operators in the eigenstates of the
integrable XXZ chain and of the quantum-chaotic model obtained by locally
perturbing the XXZ chain with a magnetic impurity. We show that, at frequencies
that are polynomially small in the system size, the behavior of the variances
of the off-diagonal matrix elements can be starkly different depending on the
operator. In the integrable model we find that, as the frequency
$\omega\rightarrow0$, the variances are either nonvanishing (generic behavior)
or vanishing (for a special class of operators). In the quantum-chaotic model,
on the other hand, we find the variances to be nonvanishing as
$\omega\rightarrow0$ and to indicate diffusive dynamics. We highlight which
properties of the matrix elements of local operators are different between the
integrable and quantum-chaotic models independently of the specific operator
selected.
