Near invariance of quasi-energy spectrum of Floquet Hamiltonians

The spectral analysis of the unitary monodromy operator, associated with a time-periodically (parametrically) forced Schrödinger 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_{\mathrm{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_{\mathrm{per}}\varepsilon^{-a}] . We consider H^{0} which is spatially periodic on \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 spectral near-invariance property for the monodromy operator, when acting on data which are \varepsilon -localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by -emphband-limited (Floquet–Bloch) wavepackets; (iii) 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.