This may be an obvious and silly question, but I'll ask anyway. I was wondering whether a spectrum of a self-adjoint operator with non-trivial AC part can be totally disconnected.
I know that the spectrum having measure $0$ implies no AC part. But does the spectrum being totally disconnected imply only possibly singular continuous component? From a somewhat short search, I could not find such examples. I assume however that this question either has famous counterexamples or a relatively simple proof of the statement.