  We investigate how the observable relaxation behavior of an isolated quantum
many-body system is modified in response to weak-to-moderate perturbations
within a nonperturbative typicality framework. A key role is played by the
so-called perturbation profile, which characterizes the dependence of the
perturbation matrix elements in the eigenbasis of the unperturbed Hamiltonian
on the difference of the corresponding energy eigenvalues. In particular, a
banded matrix structure is quantitatively captured by a perturbation profile
which approaches zero for large energy differences. The temporal modification
of the relaxation is linked to the perturbation profile via a nonlinear
integral equation, which admits approximate analytical solutions for
sufficiently weak and strong perturbations, and for which we work out a
numerical solution scheme in the general case. As an example, we consider a
spin lattice model with a pronounced banded matrix structure, and we find very
good agreement of the numerics with our analytical predictions without any free
fit parameter.
