Considering a one-dimensional case, and if it has the following relation: $$(1) : [H,a]= \pm \omega a$$
then there are evenly spaced spectrum lines.
If we sink $\omega \rightarrow 0$ , or, accordingly, the potential becomes less and less curved, and eventually - flat. then we have:
the spectrum lines will theoretically get narrower and narrower, and eventually overlap, and hence degeneracy. However, it seems very strange to me:
Based on the argument, then one eigenvalue corresponds to infinite energy eigenstates for a one-dimension "oscillator" (now technically a free particle).
Really? Infinite? I think for a a free particle with certain eigenenergy, it only has two eigenstates, as it either moving right or left (no infinity, at least for the one-dimensional situation)?
I think my intuition is not quite right, but I am not sure where. (I am not even very sure if the whole argument works)