Consider an observable in quantum mechanics, with a degenerate eigenvalue in a continuous spectrum.
Is it possible for such an eigenvalue to have a finite degeneracy?
If the degeneracy is infinite, can it have countably infinite eigenvectors? (that is, can its eigenvectors be listed?)
Now suppose we have a degenerate eigenvalue in a discrete spectrum.
Is it possible for such an eigenvalue to be infinitely degenerate? If so, are the corresponding eigenvectors countable or uncountable?
I am also interested in how you would write the unit operator (the completeness relation) in each of these cases.