Theorem: Unitary passive evolution and transfer kernel

Equation (eq:passiveODE) defines a unitary map

from initial memory and incoming port to final memory and outgoing port. Its memory annihilator is

The switching is smooth and finite, and

Proof. The energy identity

follows by substitution. Integration proves that the map is isometric. Given final memory and outgoing amplitude, solve backwards the equation obtained by substituting $u=u_{\mathrm{out}}+\overline{\lambda }z$; bounded coefficients give a unique initial memory and incoming amplitude. Thus the isometry is onto.

The memory solution is

Since $D^{\prime }=\kappa D$, $D(a)=1-\tau$, and $D(b)=1$,

The transfer kernel is therefore $\sqrt{\tau }\overline{f(t)}$. Second quantization gives (eq:captureann). The integral and peak bound follow from the same identities. Because $f$ is smooth and supported away from the endpoints and $D\ge 1-\tau >0$, the switched coefficient $\lambda$ is smooth.