B2 - degenerate eigenstates

in the degenerate substate, the correct eigenstates will be $s$ orthonormal eigenstates $| ϕ_{n α}^{0} ⟩$ , where $α = 1, \dots, s$ , and $⟨ ϕ_{n p}^{0} | ϕ_{n q}^{0} = δ_{p} q ⟩$ , and ${\hat{H}}_{0} | ϕ_{n i}^{0} ⟩ = E_{n}^{0} | ϕ_{n i}^{0} ⟩$ , spanning the subspace of dimension $s$ , $D_{s}$
the total space is the direct sum: $D_{s} \oplus D$ , where $D$ is spanned by the eigenstates of ${\hat{H}}_{0}$ not in $D_{s}$
these are the other $| ϕ_{m}^{0} ⟩$ eigenstates that may include non-degenerate and other degenerates sets where $⟨ ϕ_{m}^{0} | ϕ_{p}^{0} ⟩ = δ_{m p}$ and $⟨ ϕ_{m}^{0} | ϕ_{n i}^{0} ⟩ = 0$
zeroth order:

({\hat{H}}_{0} - E_{n}^{0}) | ϕ_{n i}^{0} ⟩ = 0

({\hat{H}}_{0} - E_{n}^{0}) | ϕ_{n i}^{1} ⟩ = (E_{n i}^{1} - \hat{V}) | ϕ_{n i}^{0} ⟩

({\hat{H}}_{0} - E_{n}^{0}) | ϕ_{n i}^{2} ⟩ = (E_{n i}^{1} - \hat{V}) | ϕ_{n i}^{1} ⟩ + E_{n i}^{2} | ϕ_{n i}^{0} ⟩

note that $| ϕ_{n i}^{1} ⟩, | ϕ_{n i}^{2} ⟩$ aren't generally eigenstates of ${\hat{H}}_{0}$
acting $⟨ ϕ_{n j}^{0} |$ (degenerate with $| ϕ_{n i}^{0} ⟩$ ) on the first-order term:

⟨ ϕ_{n j}^{0} | ({\hat{H}}_{0} - E_{n}^{0}) | ϕ_{n i}^{1} ⟩ = ⟨ ϕ_{n j}^{0} | (E_{n i}^{1} - \hat{V}) | ϕ_{n i}^{0} ⟩ = 0

⟨ ϕ_{n j}^{0} | \hat{V} | ϕ_{n i}^{0} ⟩ = E_{n i}^{1} δ_{i j}, \forall i, j = 1, \dots, s

this implies that the correct eigenstates mst be chosen to make the $s \times s$ matrix diagonal in $D_{s}$
with $i = j$ gives the first-order shifts:

⟨ ϕ_{n j}^{0} | \hat{V} | ϕ_{n i}^{0} ⟩ = E_{n i}^{1}