Within first order (or linear order) quantum perturbation theory, the Schrödinger equation (for a state $i$) can be written:
$$\delta H |\psi_i^0 \rangle + H^0 |\delta\psi_i\rangle=\delta\varepsilon_i|\psi_i^0\rangle+\varepsilon_i^0|\delta\psi_i\rangle $$
AKA Sternheimer's equation, where the exponent "$0$" denotes the ground-state, and the symbol $\delta$ corresponds to first order perturbations.
Now, I'm reading somewhere 2 things that confuse me:
$\langle \psi_j^0|\delta\psi_i\rangle = \frac{\langle \psi_j^0|\delta H|\psi_i^0\rangle - \delta \varepsilon_i\langle\psi_j^0|\psi_i^0\rangle}{\varepsilon_i^0-\varepsilon_j^0}$
$\langle \psi_j^0 | \delta\psi_i\rangle \sim \langle\psi_j^0|\delta H|\psi_i^0\rangle$
I don't understand how to obtain 1 from Sternheimer's equation, and I also don't get why the approximation 2 can be made.