Many-body perturbation theory
We can thus rewrite the exact wave function as
$$
\vert \Psi_0\rangle= (\hat{P}+\hat{Q})\vert \Psi_0\rangle=\vert \Phi_0\rangle+\hat{Q}\vert \Psi_0\rangle.
$$
Going back to the Schr\"odinger equation, we can rewrite it as, adding and a subtracting a term \( \omega \vert \Psi_0\rangle \) as
$$
\left(\omega-\hat{H}_0\right)\vert \Psi_0\rangle=\left(\omega-E+\hat{H}_I\right)\vert \Psi_0\rangle,
$$
where \( \omega \) is an energy variable to be specified later.