B1 - the degeneracy problem

suppose $E_{n}^{0}$ of ${\hat{H}}_{0}$ is $s$ -fold degenerate
there are $| u_{n α}^{0} ⟩$ linearly independent orthogonal eigenfunctions belonging to this eigenvalue, with $α = 1, \dots, s$
any linear combination of these is also an eigenfuction of ${\hat{H}}_{0}$
the second order non-degenerate perturbation theory assumed $| {\hat{V}}_{n p} | ≪ E_{p}^{0} - E_{n}^{0}$ , which does not hold for $E_{n}^{0} = E_{p}^{0}$
for zero-order states needed in the PT expansion are not uniquely defined
considering a 2-fold degeneracy with $| u_{n_{1}}^{0} ⟩, | u_{n_{2}}^{0} ⟩$ , both having $E_{n}^{0}$
the deminators in the first-order correction to wavefunction and second-order correction to energy will be zero
if a perturbation $\hat{V}$ lifts the degeneracy giving ${| v ⟩}_{n_{1}}, | v_{n_{2}} ⟩$ , whose linear combinations will not be eigenfunctions as $E_{1} \neq E_{2}$
slowly removing the perturbation $β \to 0$ , these will return to ${| u_{n_{1}}^{0} ⟩}^{'}, {| u_{n_{2}}^{0} ⟩}^{'}$ , different from the original eigenstates
so need to find $s$ normed states:

| ϕ_{n i}^{0} ⟩ = \sum_{α = 1}^{s} c_{i α} | u_{n α}^{0} ⟩, i = 1, \dots, s

$c_{i α}$ are constant coefficients which are the correct linear combinations for the PT