If A and B are self-adjoint operators, each of which possesses a complete set of eigenvectors, then AB=BA if and only if there exists a complete set of eigenvectors which are eigenvectors of both A and B.
I know how the proof goes for discrete spectrum with degeneracy accounted, but how can I prove this considering degeneracy of a continuous spectrum?
For example, in the continuous case, to prove that if A and B have a commom complete set of eigenvectors ⟹ that AB=BA, it is common, when we have no degeneracy, to write down A and B in the following way
∫da|a⟩⟨a| and ∫db|b⟩⟨b|.
How could I write that considering degeneracy?