\[\begin{split}H(t) =\begin{pmatrix}E & v(t) \\v(t) & -E\end{pmatrix} =\underbrace{\begin{pmatrix}E & 0 \\0 & E\end{pmatrix}}_{H_0} + \underbrace{\begin{pmatrix}0 & v(t) \\v(t) & -2E\end{pmatrix}}_{H'}\end{split}\]