be (possibly unbounded) skew-adjoint operators on ${\mathcal{H}}$, with domains $D(A),$ $D(B_{1}), \ldots, D(B_{p})$. Let us introduce the controlled equation $$\label{eq:main}
\frac{d\psi}{dt}(t)=(A+u_1(t)B_1+ \cdots + u_p(t) B_p) \psi(t),  \quad u(t) \in U. $$

We say that $A$ satisfies ($\mathbb{A}1$) if the following assumption is true:

($\mathbb{A}1$)

:   $A$ has discrete spectrum with infinitely many distinct eigenvalues (possibly degenerate).

Denote by $ \Phi$ a Hilbert basis $(\phi_k)_{k \in {\mathbb{N}}}$ of $\cal H$ made of eigenvectors of $A$ associated with the family of eigenvalues $(i \lambda_{k})_{k \in \mathbb{N}}$ and let $\mathcal{L}$ be the set of finite linear combinations of eigenstates, that is, $$\mathcal{L} = \bigcup_{k \in {\mathbb{N}}}  \mathrm{span}\{\phi_{1},\ldots,\phi_{k}\}.$$

We say that $(A,B_{1}, \ldots, B_{p},U,\Phi)$ satisfies $(\mathbb{A})$ if $A$ satisfies ($\mathbb{A}1$) and the following assumptions hold:

($\mathbb{A}2$)

:   $\phi_k \in D(B_{j})$ for every $k \in
    \mathbb{N}, j=1,\ldots,p $;

($\mathbb{A}3$)

:   $A+u_1B_1+ \cdots + u_p B_p:\mathcal{L} \to \mathcal{H}$ is essentially skew-adjoint for every $u\in U$.

If $(A,B_{1}, \ldots, B_{p},U,\Phi)$ satisfies $(\mathbb{A})$ then, for every $(u_{1},\ldots,u_{p}) \in U$, $A+u_{1}B_{1} + \cdots + u_{p} B_{p}$ generates a subgroup $e^{t(A+u_{1}B_{1} + \cdots + u_{p} B_{p})}$ of the group of unitary operators $\mathbf{U}({\mathcal{H}})$. It is therefore possible to define the propagator ${\Upsilon}^{u}_{T}$ at time $T$ of system  associated with a 