the discrete components of $|\Psi(t) \>$ alone. Rather than the exact equation (\[uncoup\]) in the simpler model, we will obtain a relation which is accurate to fourth order in the potentials. For this purpose we begin in straightforward fashion to integrate (\[three\]) obtaining an expression for $\s^{\a_3}$ in terms of $\phi^{\a_2}$. We then substitute this in (\[two\]) and integrate that, and substitute the result in the first equation which becomes \[mess\] [d dt]{} a\^n(t) &=& -\_[m’]{} \_0\^t dt’ e\^[iE\_[n\_2]{}t+iE\_[\_2m]{}t’]{} V\^n\_V\^\_[m’]{} a\^[m’]{}(t’)\
&+&i\_2\_3’\_2 \_[0]{}\^tt’\_[0]{}\^[t’]{} t” e\^[iE\_[n\_2]{}t+iE\_[\_2\_3]{}t’+iE\_[\_3’\_2]{}t”]{} V\^n\_[\_2]{}V\^[\_2]{}\_[\_3]{}V\^[\_3]{}\_[’\_2]{}\^[’\_2]{}(t”)

The next step is to substitute for $\phi^{\a'_2}(t'')$ in the equation above the value obtained by integrating the first term of (\[two\]) with “source” $a^m$. The term with “source” $\phi^{\a_2}$ is of higher order in the potentials through (\[three\]) and can be dropped. This gives us the net contribution of triple-trace intermediate states, denoted by \[trip\] [d dt]{} a\^n(t)\_[triple]{} &=& \_[m]{} \_2\_3’\_2 \_[0]{}\^t t’ \_[0]{}\^[t’]{} t” \_[0]{}\^[t”]{}t”’ e\^[iE\_[n\_2]{}t+iE\_[\_2\_3]{}t’]{}\
&& e\^[iE\_[\_3’\_2]{}t”+iE\_[\_2m]{}t”’]{}  V\^n\_[\_2]{}V\^[\_2]{}\_[\_3]{}V\^[\_3]{}\_[’\_2]{}V\^[’\_2]{}\_m a\^m(t”’)

To obtain the contribution of single-trace intermediate states at order $g_2^4$, we integrate the first term of (\[mess\]) to find an expression for $a^n(t)$ (with $n \to m$). This expression is then reinserted for $a^{m'}(t')$ in (\[mess\]) to obtain the iterated contribution 