In Quantum Field Theory in Curved Spacetime, are particle states energy eigenstates? 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?
 A: In the way I originally wrote the question, it was a bit difficult to see the issue without reading the paper, but I believe I have now found the issue.
The wavepackets Wald uses in the analysis are peaked around specific frequencies, but are still wavepackets. In other words, their energy is distributed closely around a specific energy, but, of course, involve more than one energy in superposition. Hence, these modes are not energy eigenstates.
This is not really an issue with the definition of particle. Firstly, notice that a one-particle state does not need to have a well-defined energy: just pick the superposition $\frac{1}{\sqrt{2}}\left(|\mathbf{p}\rangle + |\mathbf{q}\rangle\right)$, for example, and you have a one-particle state which does not have well-define momentum. The reason for the construction to use wavepackets instead of energy eigenstates is then the technical fact that the energy eigenstates would not provide normalizable states, and hence they can't really be used as a basis for the one-particle Hilbert space. Using a basis formed out of wavepackets solves this technical issue.
In short, the $F_{i1}$ modes are of purely positive frequency with respect to inertial time, but they are also a superposition of modes of different frequencies.
