Diagonalization of a Hamiltonian in a transformed reference frame

Under a time-dependent transformation $V(t)$ of the state vectors $|{psi}rangle$

begin{equation}
|psi'(t)rangle = V(t) |psi(t)rangle
end{equation}

The Hamiltonian $H(t)$ has to transform as

begin{equation}
H’ = V H V^{-1} – i hbar V dot{V}^{-1}
end{equation}

to preserve the form of the Schrödinger equation.

If the original Hamiltonian has instantaneous eigenvectors ${|n(t)rangle}_n$ of eigenvalues $E_n(t)$ such that

begin{equation}
H(t)|n(t)rangle = E_n(t) |n(t)rangle,
end{equation}

can one also find eigenvector and eigenvalues of the transformed Hamiltonian $H’$? I.e., can one always diagonalize $H’$?

I know that, in particular cases, it is possible to do that. I am wondering if there is a scheme to do it in the general case.