  We consider time-dependent perturbations which are relatively bounded with
respect to the square root of an unperturbed Hamiltonian operator, and whose
commutator with the latter is controlled by the full perturbed Hamiltonian. The
perturbation is modulated by two auxiliary parameters, one regulates its
intensity as a prefactor and the other one controls its time-scale via a
regular function, whose derivative is compactly supported in a finite interval.
We introduce a natural generalization of energy conservation in the case of
time-dependent Hamiltonians: the boundedness of the two-parameter unitary
propagator for the physical evolution with respect to the $n/2$-th power energy
norm for all $n\in\mathbb{Z}$. We provide bounds of the $n/2$-th power energy
norms, uniformly in time and in the time-scale parameter, for the unitary
propagators, generated by the time-dependent perturbed Hamiltonian and by the
unperturbed Hamiltonian in the interaction picture. The physically interesting
model of Landau-type Hamiltonians with an additional weak and
time-slowly-varying electric potential of unit drop is included in this
framework.
