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We study Schrodinger equations which are periodic in space and time. These models are inspired by recent experimental progress in the study of wave propagation and quantum mechanics under time-periodic forcing. Time-periodic Hamiltonians, however, are not as well understood as their static (autonomous) analogs.

In particular, many discrete models of periodic materials (e.g., of graphene) are known to develop spectral gaps under a time-periodic driving. In PDEs, however, no such gaps are conjectured to form. How do we reconcile these two facts? Using periodic homogenization, we prove that the driven Schrodinger equation has an “effective gap” - a new and physically-relevant relaxation of a spectral gap.

Taking a broader perspective, we ask - how does time-periodic forcing affect a general band structure? We will show that a spectrally-local notion of stability can be formulated and proved, in a way which again corresponds with periodic homogenization theory.