\d^m_n$. The time integrals are then easily done, but separately for the two cases $m \ne n$ and $m = n$. The contributions of the various terms to $\frac{d}{dt} a^n(t)$ at small $t$ are then analyzed as above. In each case there is one term which contributes at $t=0$ via the $i\e$ prescription. In each there is a $\d(E_{n\a_2})$ which drops because $V^n_{\a_2}=0$. The contributions of two (non-singular) principal value integrals remain in the final result \[sing1\] a\^n(0)\_[single]{} &=&i\_[m]{} U\^n\_m U\^m\_n\
&& -i U\^n\_n d\_2 which is largely written in terms of the composite interaction \[mek\] U\^n\_m =d\_2

We can now interpret the results (\[trip1\],\[sing1\]) in terms of (\[toy\]). We find the decay width \[wid\] \_n = 2d\_3 (E\_[\_3 n]{}) U\^n\_[\_3]{}U\^[\_3]{}\_n and energy shift \[eshif\] ø\_n &=& U\^n\_n - \_[m]{} U\^n\_m U\^m\_n\
&& + U\^n\_n d\_2\
&& -d\_3 U\^n\_[\_3]{}() U\^[\_3]{}\_n

We are primarily interested in the decay width, which will be evaluated for the BMN system (\[heff\], \[Vcoef\]) in section 5. However, we note that the energy shift agrees with the result of fourth order non-degenerate (time-independent) perturbation theory for any quantum-mechanical system whose state space and interaction structure are the same as the present model as defined in (\[int\]) 