RAGE theorem for absolutely continuous spectrum?

Question

Given a (possibly unbounded) self adjoint operator $A$ acting on a Hilbert space $\mathcal{H}$, the RAGE theorem gives a characterisation of $$ \|P_{c}\psi\|^2 $$ where $\psi\in\mathcal{H}$ and $P_c$ denotes the orthogonal projection onto the subspace associated with the continuous spectrum of $A$. I was wondering whether a similar thing can be done for the absolutely continuous part (w.r.t. Lebesgue measure) or whether there are any dynamical characterisations of the absolutely continuous spectrum?

Answer 1

I myself am not aware of a dynamical characterization of absolutely or singular continuous spectra with the same generality as the RAGE theorem.

If you are willing to restrict the scope to quantum mechanics and the theory of Schrödinger operators, however, you get sort of a dynamical characterization of absolutely continuous spectrum: establishing absence of singular continuous spectrum is related to asymptotic completeness, i.e. the density of scattering states plus bound states. See e.g. Reed, Simon, Vol. 4, Sec. XIII.6.
This is of course not on the same level of generality as RAGE, because you require a free time evolution to compare to.