I was reading a paper today claiming observation of a new $2^+$ state in $^{12}\mbox{C}$, which would correspond to a sort of rotationally excited Hoyle state. Looking at NuDat reveals that this region of excitation energy seems well probed, with many states known above and below the new state. This got me to wondering:
What fraction of excited states are known in the light nuclei? (Presumably, heavier nuclei have very many states and must be less completely known. Also, let's assume we stay below the single-nucleon ejection energy so that we have a definite upper bound.)
Are there some regions where we know that all excited states are known? (Probably "yes" for systems such as the deuteron: there aren't any.)
Are there enough too-broad states such that we can never really know if we've got them all? (This new state is 0.8 MeV wide at 10.0 MeV above ground.)