Santoro's proof of quantum floquet theorem

Question

I'm following Santoro's lecture notes, https://www.ggi.infn.it/sft/SFT_2019/LectureNotes/Santoro.pdf, and I'm having trouble seeing how he reaches 2.17. I understand that we can write $$F_0 |u_j(t_0)\rangle = e^{-i\epsilon_j T}|u_j(t_0)\rangle,$$ but I don't see how this implies the existence of an operator $H_{F_0}$ such that $$e^{-iH_{F_0}T}|u_j(t_0)\rangle = e^{-i\epsilon_j T}|u_j (t_0)\rangle.$$ How do we know that such an operator exists?

Answer 1

It's been a while and I think I understand now. The eigenvalues all have modulus 1 implying that the operator $F_0$ is unitary. This also applies to its generator. We also have that $F_0 \rightarrow 1$ in the limit $T \rightarrow 0$. Furthermore $$F_0 (T_2) F_0 (T_1) |u_j (t_0) \rangle = e^{-i \epsilon_j (T_1 + T_2)} |u_j (t_0) \rangle,$$ so the generator must be additive.

Combining the above, we find a generator $1 - i H_{F_0} t$. If we take $t = T/N$ and apply $N$ successive operations, we get $\left(1 - \frac{i H_{F_0} T}{N} \right)^N$. Taking the limit $$\lim_{N \rightarrow \infty} \left( 1 - \frac{i H_{F_0} T}{N} \right)^N = e^{-i H_{F_0} T}.$$

Feel free to correct me.