\[sing\] [d dt]{} a\^n(t)\_[single]{} &=& \_[m,m’]{} \_2’\_2 \_[0]{}\^t t’ \_[0]{}\^[t’]{} t” \_[0]{}\^[t”]{} t”’e\^[iE\_[n\_2]{}t+iE\_[\_2m]{}t’]{}\
&&e\^[iE\_[m’\_2]{}t”+iE\_[\_2m’]{}t”’]{}  V\^n\_[\_2]{}V\^[\_2]{}\_[m]{}V\^[m]{}\_[’\_2]{}V\^[’\_2]{}\_[m’]{} a\^[m’]{}(t”’) The full expression for $\frac{d}{dt}a^n(t)$ to fourth order is \[full\] [d dt]{} a\^n(t) &=& -\_2 \_[m’]{} \_0\^t dt’ e\^[iE\_[n\_2]{}t+iE\_[\_2m]{}t’]{} V\^n\_[\_2]{} V\^[\_2]{}\_n\
&&+  a\^n(t)\_[triple]{}  + a\^n(t)\_[single]{} The first term is just the short-time approximation to (\[unk\]) in the simple model in different notation. We will not discuss it further since it does not contribute to the fourth order amplitudes of primary concern in the generalized model.

We now make the short-time approximation $a^n(t') \approx 1,~~
a^m(t') \approx 0$ for $m \ne n$ in (\[trip\]) and (\[sing\]). We will present the treatment of the triple-trace part explicitly and then summarize the rather similar steps needed for the single-trace part.

The nested set of time integrals in (\[trip\]) can easily be done, and the result (including the overall factor $\exp i(E_{n\a_2})t)$ is \[tint\] - + + In an obvious fashion we subtract and add $1$ in the last two terms of (\[tint\]). We thus obtain six terms with the structure $\exp iEt~-1$ divided by energy denominators. In our discussion of the contribution of these six terms to (\[trip\]) in the short-time limit, we need the fact that 