It generally depends on the time $t$ explicitly[^6] and it is supposed to be a Hermitian operator, i.e.  ${\mathcal{H}}^\dag(t)={\mathcal{H}}(t)$. The Schrödinger equation (see [@Dirac-PQM § 27] or [@Prugovecki-QMinHS chapter V, Sec. 3.1]) $$\label{2.5}
{\mathrm{i}\hbar}\frac{d \psi(t)}{d t} = {\mathcal{H}}(t) \psi(t),$$ with ${\mathrm{i}}\in\mathbb{C}$ and $\hbar$ being respectively the imaginary unit and the Plank’s constant (divided by $2\pi$), together with some initial condition $$\label{2.5a}
\psi(t_0)=\psi_0\in{\mathcal{F}}$$ is postulated in the quantum mechanics

The substitution of (\[2.1\]) into (\[2.5\]) shows that there is a 1:1 correspondence between ${\mathcal{U}}$ and ${\mathcal{H}}$ described by $$\label{2.6}
{\mathrm{i}\hbar}\frac{\partial {\mathcal{U}}(t,t_0)}{\partial t} = {\mathcal{H}}(t)\circ{\mathcal{U}}(t,t_0),
\qquad
{\mathcal{U}}(t_0,t_0) = {\mspace{2mu}\mathsf{i}\mspace{-1mu}\mathsf{d}}_{\mathcal{F}}$$ where $\circ$ denotes composition of maps. If ${\mathcal{U}}$ is given, then $$\label{2.7}
{\mathcal{H}}(t) =
{\mathrm{i}\hbar}\frac{\partial {\mathcal{U}}(t,t_0)}{\partial t} \circ {\mathcal{U}}^{-1}(t,t_0) =
{\mathrm{i}\hbar}\frac{\partial {\mathcal{U}}(t,t_0)}{\partial t} \circ {\mathcal{U}}(t_0,t),$$ where we have used the equality $${\mathcal{U}}^{-1}(t_2,t_1) = {\mathcal{U}}(t_1,t_2)$$ which follows from (\[2.1\]) (see also below (\[2.8\]) or Sect. \[IV\]). Conversely, if ${\mathcal{H}}$ is given, then [@Messiah-QM chapter VIII, § 8] ${\mathcal{U}}$ is the unique solution of the integral equation $
{\mathcal{U}}(t,t_0) = {\mspace{2mu}\mathsf{i}\mspace{-1mu}\mathsf{d}}_{\mathcal{F}}+
{{\frac{1}{{\mathrm{i}\hbar}}}} \int\limits_{t_0}^{t} {\mathcal{H}}(\tau){\mathcal{U}}(\tau,t_0) d\tau,
$ i.e.  we have $$\label{2.8}
 {\mathcal{U}}(t,t_0) = \operatorname{Texp}\int\limits_{t_0}^{t}
{{\frac{1}{{\mathrm{i}\hbar}}}} {\mathcal{H}}(\tau) d\tau,$$ where $\operatorname{Texp}\int\limits_{t_0}^{t}\cdots d\tau$ is the chronological (called also T-ordered, P-ordered or path-ordered) exponent (defined, e.g as the unique solution 