Proposition: Exact switching-work ledger

The control work in this prescribed driving model is

This is a finite excess-energy statement. It is not a black-hole mass-loss law or an autonomous source-energy conservation theorem.

Proof. The reduced mean solves $\mathsf{D}{m}_s=-sgf$. Differentiating its classical energy and integrating by parts gives

It has zero driven energy before switching. Integrate in time. Since $f$ vanishes at the temporal endpoints, integration by parts gives the first equality.

In the Schrödinger description the explicitly time-dependent interaction is $g{Z}_S\int f(t,x)\Psi (x)\mathrm{d}x$. Its explicit derivative has expectation $g\int ({\partial }_{t}f)⟨Z_S\Psi ⟩\mathrm{d}x$. The conditional means give $⟨Z_S\Psi ⟩=-g{v}_{\mathrm{R}}$, independently of the source populations. Thus its work is the displayed integral.

After switching, mode expansion of the real driven solution, or the equivalent displaced mean-amplitude calculation, gives $E_{\mathrm{cl}}=\int \omega g^2{\sum }_j|k_{F,j}{|}^2\mathrm{d}\omega$. This is $\delta E_{\mathrm{tot}}$.