Degenerate perturbation theory

First-order degenerate perturbation theory diagonalizes the perturbing operator within each degenerate eigenspace of the unperturbed Hamiltonian.

If the perturbation is already diagonal in a basis of a degenerate eigenspace, those basis vectors receive its diagonal entries as first-order shifts.

When a perturbation preserves each unperturbed degenerate block and has no inter-block coupling, diagonalizing each block can give the exact spectrum.

For oscillators of frequencies $1$ and $2$, the perturbationpreserves $N_A+2N_B$. On the energy-$9/2$ basis $|3,0⟩,|1,1⟩$, it is the matrixso the symmetric and antisymmetric combinations are exact eigenstates with shifts $\pm \sqrt{6}\lambda$.