Periodic finite dynamics, quasienergy and controlled truncation — Examples

Put $F(t)=E_{0}t+(A\mathrm{/}\omega )\sin (\omega t)$. By [F1] and [F4], $F^{\mathrm{\prime }}(t)=E_0+A\cos (\omega t)$. The diagonal entries of $\exp [-iF(t){\sigma }_z\mathrm{/}\mathrm{\hslash }]$ are $e^{-iF(t)\mathrm{/}\mathrm{\hslash }}$ and $e^{iF(t)\mathrm{/}\mathrm{\hslash }}$; differentiating them using [F2] gives $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,0)=F^{\mathrm{\prime }}(t){\sigma }_{z}U(t,0)=H(t)U(t,0)$. At $t=0$ the exponential is $I_2$, so uniqueness in [F2] identifies it with the propagator.

The finite exponential group law in [F2] splits $U(t,0)=e^{-iA\sin (\omega t){\sigma }_z\mathrm{/}(\mathrm{\hslash }\omega )}{e}^{-it{E}_0{\sigma }_z\mathrm{/}\mathrm{\hslash }}$. Thus $H_F=E_0{\sigma }_z$ and the stated $P(t)$ give the required factorization. The matrix exponential is unitary by [F2], and $P(t+T)=P(t)$ by the $2\pi$ period in [F1]. Hence this is a Floquet normal form with quasienergies $E_0$ and $-E_0$, each understood modulo $\mathrm{\hslash }\omega$.

Take the constant zero Hamiltonian $H(t)=0$ on $\mathbb{C}$. Its propagator is $U(t,0)=1$ by [F3]. For each integer $n$, the Hermitian scalar $H_F^{(n)}=n\mathrm{\hslash }\omega$ satisfies $e^{-iT{H}_F^{(n)}\mathrm{/}\mathrm{\hslash }}=e^{-i2\pi n}=1=U(T,0)$, so each is a valid logarithm of the same monodromy.

In dimension one the only eigenvalue of $H_F^{(n)}$, and hence its least quasienergy representative, is $n\mathrm{\hslash }\omega$. It varies with $n$, while all representatives have the same class modulo $\mathrm{\hslash }\omega$ by [F1, F2]. Therefore the period unitary determines a quasienergy class but no branch-independent least real quasienergy.