the initial conditions above and substitute the resulting expression for $ \phi^\a(t)$ in the first equation, obtaining an equation involving only the discrete components, namely \[uncoup\] [d dt]{} a\^n(t) = -d\_[m]{} \_0\^t dt’ i(E\_n-E\_)t i(E\_-E\_m)t’ V\^n\_V\^\_m a\^m(t’) Next we separate the energy variable in the continuum measure by writing $d\a =dE d\b \r(\b,E)$ and define the matrix \[kdef\] K\^n\_m(E) = d(,E) V\^n\_V\^\_m. We assume that $K^n_m(E)$ is a slowly varying function of energy whose scale of variation is $\D E$. Substituting (\[kdef\]) in (\[uncoup\]), we find \[unk\] [d dt]{} a\^n(t) = -\_[0]{}\^ dE K\^n\_m(E)  i(E\_n-E\_)t \_[0]{}\^[t]{} dt’ i(E\_-E\_m)t’ a\^m(t’).

We now follow [@cotan] and make the short-time approximation $a^n(t') \approx 1,~~ a^m(t') \approx 0$ for $m \ne n$ in (\[unk\]). We encounter the well known integral \[wno\] \_[0]{}\^[t]{} dt’ i(E\_n-E)(t-t’) &=&\
&& (E\_n-E) + i([ 1 E\_n-E]{}). The last result is strictly correct in the limit $t \to \infty$, but it is effectively valid within integrals of functions f(E) for times much larger than the inverse scale of variation $\D E$.

This approximate treatment of (\[unk\]) thus leads to the result \[toy\] a\^n(t) && 1 - ( + iø\_n)t\
\_n &=& 2 K\^n\_n(E=E\_n)\
ø\_n &=& \_[0]{}\^ dE [K\^n\_n(E) E\_n-E]{}. The 