I was just reading this paper here: on PDE control.
They start with a PDE: $$i \partial_t \psi(t,x) = (H_0- \mu(t))\psi(t,x)$$ which is their equation (1.1).
Then in (2.1) they say that by substituting $\psi(t,x)=\psi_1(t,x)+\psi_2(t,x)$ and(!) $\psi=(\psi_1,\psi_2)$ this PDE can be rewritten as the following systems
$$i \partial_t \psi_1(t,x) = (H_0- \mu(t))\psi_2(t,x)$$ $$i \partial_t \psi_2(t,x) = -(H_0- \mu(t))\psi_1(t,x).$$
I mean somehow it is clear that $\psi$ cannot be the sum of these two and the vector at the same time, but I thought that maybe somebody here understands what they might be doing from context which is why I am asking.
It's probably
$\psi(x,t) = \psi_1(x,t) + i \psi_2(x,t)$