Theorem: Simultaneous capture from one port

Equation (eq:vectorODE) is unitary from ${\mathbb{C}}^2\oplus L^2([a,b])$ to itself. Its final memory annihilators obey

The interaction is smooth and finitely switched, and

Its resources satisfy

Proof. Differentiating $D=G^2$ gives $\dot{D}=\dot{G}G+G\dot{G}=\tau q{q}^{†}$. It follows that

Consequently

Integration proves isometry. Substituting $u=u_{\mathrm{out}}+{\lambda }^{†}z$ gives a bounded backward linear equation with a unique solution from the final memory and outgoing amplitude. The isometry is therefore onto.

The final-time fundamental solution is $G(t)$, since $\dot{G}=-GC$ and $G(b)=I$. Thus

Second quantization proves (eq:capturepair).

The integrated coupling follows from

The peak bound uses $D⪰(1-\tau )I$. The Sylvester equation for $\dot{G}$ has the integral solution

Hence $\Vert \dot{G}\Vert \le \Vert \dot{D}\Vert /(2\sqrt{1-\tau })$, and $\Vert C\Vert \le \tau \Vert q{\Vert }^2/(2(1-\tau ))$. The same bounds $\Vert \Omega \Vert$. Smoothness and finite switching follow from the supports of $f_i$ and the positive lower bound on $D$.