1$ for $i=i_0$ and $a^i(0)=0$ for $i
\ne i_0$. From the Schrödinger equation $i \frac{d}{dt}|\Psi(t)\>
= H |\Psi(t)\>$ we easily derive the component evolution \[comp1\] i a\^i(t) = H\^i\_j a\^j(t).

In string basis we would instead be interested in the state \[state2\] |(t) = \_[i]{} [a]{}\^i(t) \_i which evolves in time by the Hamiltonian ${\tilde H}$ with the initial conditions ${\tilde a}^i(0) = 1$ for $i=i_0$ and ${\tilde a}^i(0)=0$ for $i \ne i_0$.

We now attempt to relate the time-dependent expansion coefficients $a^i(t)$ and ${\tilde a}^i(t)$. The Schrödinger equation $i \frac{d}{dt}|{\tilde \Psi}(t)\> = {\tilde H} |{\tilde
\Psi}(t)\>$ can be expanded as \[sch2\] i \_[i]{} [a]{}\^i(t) S\^[-1]{} \_i = \_[m]{} [a]{}\^i(t) [H]{}S\^[-1]{} \_i. We apply $S$ to both sides and obtain the component equations \[comp2\] i \_[i]{} [a]{}\^i(t) &=& (S[H]{} S\^[-1]{})\^i\_j [a]{}\^j(t)\
&=& (S\^2 H S\^[-2]{})\^i\_j \^j(t) We now consult Sec. 4 of [@sv] and learn that $S^2 H S^{-2} =
H^\dagger$. Thus the string basis components evolve via \[comp3\] i \_[i]{} [a]{}\^i(t) = (H\^)\^i\_j [a]{}\^j(t), a curious and useful fact!

To apply this fact we simply go back to the expressions for the decay amplitude in (\[tricon\]  -  \[eshif\]) and observe that these results remain unchanged if we replace every 