Many-body perturbation theory
These operators commute meaning that
$$
\hat{Q}\frac{1}{\left(\omega-\hat{H}_0\right)}\hat{Q}=\hat{Q}\frac{1}{\left(\omega-\hat{H}_0\right)}=\frac{\hat{Q}}{\left(\omega-\hat{H}_0\right)}.
$$
With these definitions we can in turn define the wave function as
$$
\vert \Psi_0\rangle=\vert \Phi_0\rangle+\frac{\hat{Q}}{\omega-\hat{H}_0}\left(\omega-E+\hat{H}_I\right)\vert \Psi_0\rangle.
$$
This equation is again nothing but a formal rewrite of Schr\"odinger's equation
and does not represent a practical calculational scheme.
It is a non-linear equation in two unknown quantities, the energy \( E \) and the exact
wave function \( \vert \Psi_0\rangle \). We can however start with a guess for \( \vert \Psi_0\rangle \) on the right hand side of the last equation.