Why do we need the extended Floquet Hilbert Space $\mathcal{F}$ to study the Time Periodic Hamiltonian (i.e., $H(t+T)=H(t)$)? What is the problem with the Normal Hilbert Space?
Where we define the Floquet Hilbert space as $\mathcal{F}$=$\mathcal{H}$ $\otimes$$\mathcal{T}$, where $\mathcal{T}$ is a space of bounded periodic function over $[0,T]$ where $T$ is the time period of the Hamiltonian.
Reference: https://arxiv.org/pdf/1805.01190.pdf
$\newcommand{\drangle}{\rangle\hspace{-2pt}\rangle} \newcommand{\dlangle}{\langle\hspace{-2pt}\langle}$ It is a mathematical trick which allows us to build a Hilbert space on which we can package the Hamiltonian operator at different times into one operator, for example to apply the spectral theorem and obtain the Floquet modes. The usual Hilbert space would be too small for this. Let us see how this work in details :
On $\mathcal F = \mathcal H \otimes \mathcal T$ (with $\mathcal T$ the space of square integrable $T$-periodic functions), we can define : \begin{align} \tilde H &= \int_0^T \frac{dt}{T} H(t)\otimes |t)(t|\\ \tilde Z &= I_\mathcal H \otimes\int_0^T\frac{dt}{T} |t)i\frac{d}{dt}(t| \end{align}
Both operators are Hermitian, therefore so is the difference $\tilde H - \tilde Z$. Therefore, there is a basis of $\mathcal F$ consisting of eigenvectors of $\tilde H - \tilde Z$.
Actually, as the operators : $$\tilde\mu_n = I_\mathcal H\otimes \int_0^T |t)e^{in\Omega t}(t| \frac{\text dt}T$$ satisfy $[\tilde \mu_n , \tilde H - \tilde Z] = n\tilde \mu_n$ and $\tilde \mu_n\tilde \mu_n = \tilde \mu_{n+m}$, we can generate all eigenvectors of $\tilde H - \tilde Z$ by applying $\tilde \mu_n$ to the eigenvectors with eigenvalues in the Floquet zone $\epsilon \in [-\Omega/2,\Omega/2)$. Such an eigenvector is precisely a Floquet mode.
To recap, the extended Hilbert space allows us to consider the LHS of the Floquet-Schrödinger equation : $$(H(t) -i\frac{\text d}{\text dt})|\phi(t)\rangle = \epsilon|\phi(t)\rangle$$ at all times $t\in[0,T)$ as one hermitian operator. The spectral theorem then ensures the existence of the Floquet modes.
