quantities $ \G_n, ~\D \o_n$ are interpreted as the decay width and energy shift of the unstable state to lowest order in the interaction $V$. The approximations made in the derivation are valid under the conditions: $1/\D E << t << 1/\G_n$, the time t must be long compared to the inverse scale of variation of $K^n_n(E)$ and sufficiently short to justify the short-time approximation to (\[unk\]). Note that it was not necessary to specify a scalar product in Hilbert space. It is necessary to define the measure $d\a =dE d\b \r(\b,E)$ explicitly. In our problem this measure is determined by the BMN limit of the discrete formulation.

There are additional checks of the self-consistency of the method. One can show that the components $a^m(t),~~m \ne n$ satisfy $a^m(t) \sim t^2$ for small $t$, and that the unitarity constraint $|a^n|^2 + \int d\a |\phi^\a| = 1$ is satisfied to order $t$.

Applied to our problem the treatment above gives the result $\G_n=0$, since $K^n_n(E_n)$ vanishes at degeneracy[^4]. This is because triple-trace states enter the dynamics only at higher order in the coupling $g_2$. The toy model above must be generalized to include this effect. Note that the energy shift $ 