I've been reading the paper Particle and energy cost of entanglement of Hawking radiation with the final vacuum state, by R. M. Wald (arXiv: 1908.06363 [gr-qc], DOI: 10.1103/PhysRevD.100.065019). It involves studying a two-dimensional moving mirror spacetime which undergoes acceleration as to emit Hawking-like radiation and then return to inertial motion. The goal is to understand a few aspects of Hawking radiation, black hole evaporation, and the information paradox.
In the process of analysis, more specifically on Sec. VI, Wald shows that there are modes of purely positive frequency with respect to inertial time emitted at late times (those are defined on Eq. (24)). These, as far as I understand, can be interpreted as inertial particles being emitted at late times.
A bit after, in the paragraph right before Eq. (28), Wald states that
Since the mode $F_{i1}$ associated with the Hawking mode $h_i$ is not an eigenstate of inertial energy and since the different $F_{i1}$ modes may overlap at $\mathcal{I}^+$, we do not know of any simple way [...]
I find this affirmation a bit odd: the modes $F_{i1}$ are precisely what he just defined to be inertial particles at late times. While I guess it makes sense to me that these modes do not need to be energy eigenstates at all times due to the acceleration of the mirror it seems that at late times (the case of interest) they should be energy eigenstates, since the mirror is already in inertial motion. In other words, the region of interest of spacetime is diffeomorphic to some auxiliary stationary spacetime in which the states should be energy eigenstates.
My question then is: when dealing with Quantum Field Theory in Curved Spacetimes, are particle states defined in terms of some Killing vector field $\xi^a$ energy eigenstates for the notion of energy defined by such Killing field? If they are, then how come Wald's affirmation? If they are not, why?