For this question I was looking for the Floquet states of a quantum harmonic oscillator driven by a non-resonant harmonic force, and I had a rather harder time finding it than the simplicity of the question would suggest. I'm sure the result exists, but it's somewhat buried under a mound of papers dealing with damped oscillators (which are of course interesting, including many real, useful applications in open quantum systems, but the undamped oscillator is also important) and many papers exploring formalisms which are not necessarily that crucial to the result. I would therefore like to ask it here so hopefully there's an easier waymarker for future askers of this question.
It seems, moreover, that these states are not as easy to get at. For instance, the paper
Squeezing in Floquet States and Quasi-energies of Harmonic Oscillator Driven by a Strong Periodic Field. M. Janati-Idrissi et al. African Journal Of Mathematical Physics 10, 21-30 (2011).
is quite recent, and it only goes to second order in the driving. This suggests that maybe these states are not actually known in exact form. In addition, it seems (Commun. Math. Phys. 215, 245 (2000)) that the Floquet quasienergy spectrum is rather different than from the static case (purely absolutely continuous instead of discrete) so there's more at play, with some disagreements in recent literature.
So: Are there known, exact expressions for the Floquet states of a periodically-forced, undamped harmonic oscillator? I'm looking for explicit expressions for a complete, orthogonal set of Floquet states, as well as a characterization of the quasienergy spectrum, and an explanation of any significant differences it has with respect to the unperturbed oscillator and the perturbative solutions.