  The spectral analysis of the unitary monodromy operator, associated with a
time-periodically (paramatrically) forced Schrodinger equation, is a question
of longstanding interest. Here, we consider this question for Hamiltonians of
the form
  $$H^{\varepsilon}(t)=H^0 + \varepsilon^a W(\varepsilon^a t, -i\nabla)\, ,$$
where $H^0$ is an unperturbed autonomous Hamiltonian, $a\geq 1$, and
$W(T,\cdot)$ has a period of $T_{\rm per} >0$. In particular, in the small
$\varepsilon>0$ regime, we seek a comparison between the spectral properties of
the monodromy operator, the one-period flow map associated with the
$H^\varepsilon(t)$ dynamics, and that of the autonomous (unforced) flow,
$\exp[-iH^0 T_{\rm per} \varepsilon ^{-a}]$. We consider $H^0$ which is
spatially periodic on $\mathbb{R} ^n$ with respect to a lattice. Using the
decomposition of $H^0$ and $H^\varepsilon(t)$ into their actions on spaces
(Floquet-Bloch fibers) of pseudo-periodic functions, we establish a near
spectral-invariance property for the monodromy operator, when acting data which
are $\varepsilon$-localized in energy and quasi-momentum. Our analysis requires
the following steps: (i) spectrally-localized data are approximated by {\it
band-limited (Floquet-Bloch) wavepackets}; (ii) the envelope dynamics of such
wavepackets is well approximated by an effective (homogenized) PDE, and (iii)
an exact invariance property for band-limited Floquet-Bloch wavepackets, which
follows from the effective dynamics. We apply our general results to a number
of periodic Hamiltonians, $H^0$, of interest in the study of photonic and
quantum materials.
