  We identify constraints in the energy spectra of quantum theories that have a
global $O(N)$ symmetry, where $N$ is treated as a continuous parameter. We
point out that a class of evanescent states fall out of the spectrum at integer
values of $N$ in pairs, via an annihilation mechanism. This forces the energies
of the states in such a pair to approach equality as $N$ approaches a certain
integer, with both states disappearing at precisely integer $N$ and the point
of would-be degeneracy. These constraints occur between different irreducible
representations of the analytic continuation of $O(N)$ and hold
non-perturbatively. We give examples in the spectra of the critical $O(N)$
model.
