# Time-reversal and its effect in the eigenfunctions of the Floquet operator

When time-reversal is present in a kicked Hamiltonian system (e.g. kicked rotator) it can be shown that there exists a basis in which the associated Floquet operator $$U$$ is symmetric. One can then try to find the eigenvalues and eigenvectors of this operator

$$U |v\rangle = e^{i\phi}|v\rangle.$$

If time-reversal holds, is it possible to classify the eigenfunctions in muttually exclusive sets? By this I mean the following: for example when parity is present in Hamiltonian system, then for an eigenfunction $$|w\rangle$$ of the Hamiltonian $$H$$ one can classify the eigenfunctions as even or odd according to the parity operator $$P$$, i.e.

$$P|w\rangle = \pm |w\rangle.$$

Can something similar exists for time-reversal?

There are two types of time reversal invariance. One leads to the Floquet operator $$U$$ being a symmetric unitary matrix, and it will have real eigenvectors. The other leads to a different symmetry in $$U$$ that has eigenvectors that appear in time-reversal pairs. This means one can find a Floquet Hamiltonian (a static Hamiltonian) that has the correct time reversal symmetry.