Theorem: No exact finite-time transfer

For finite $T$, $t_0\ge 0$, and bounded piecewise continuous real controls in the stated one-species model, $|B(T)|<1$.

Proof. If $|B(T)|=1$, unitarity gives $\psi (T)=0$. Since $f(\omega )>0$, the packet formula implies

for almost every positive $\omega$. Both sides extend to entire functions, so equality holds everywhere. At $\omega =\mathrm{i}y$, the left side is $e^{y{t}_0}$. For $t_0>0$ it contradicts boundedness of the right side. For $t_0=0$, dominated convergence makes the right side tend to zero while the left side is one.