Rotating-wave finite-time error

[Exact Duhamel identity with units restored] Suppose first $s<t$. Put $F(r)=U(t,r)U_{\mathrm{a}\mathrm{v}\mathrm{g}}(r,s)$ on $[s,t]$. By [F1]–[F2], $F$ is norm-$C^1$ and $F^{\mathrm{\prime }}=(i\mathrm{/}\mathrm{\hslash })U(t,r)(H(r)-A)U_{\mathrm{a}\mathrm{v}\mathrm{g}}(r,s)=(i\mathrm{/}\mathrm{\hslash })U(t,r)V(r)U_{\mathrm{a}\mathrm{v}\mathrm{g}}(r,s)$. Its endpoint values are $F(t)=U_{\mathrm{a}\mathrm{v}\mathrm{g}}(t,s)$ and $F(s)=U(t,s)$. The FTC [F3] therefore gives $U(t,s)-U_{\mathrm{a}\mathrm{v}\mathrm{g}}(t,s)=-(i\mathrm{/}\mathrm{\hslash }){\int }_s^{t}U(t,r)V(r)U_{\mathrm{a}\mathrm{v}\mathrm{g}}(r,s)dr$. This identity holds as an operator-norm Bochner integral, not merely on a chosen finite-dimensional subspace.

[Integration by parts removes only the middle derivative] Let $P(r)=U(t,r)$, $Q(r)=U_{\mathrm{a}\mathrm{v}\mathrm{g}}(r,s)$ and $G(r)=P(r)Z(r)Q(r)$. All three factors are norm-$C^1$. The exact product rule is $G^{\mathrm{\prime }}=P^{\mathrm{\prime }}ZQ+P{Z}^{\mathrm{\prime }}Q+PZ{Q}^{\mathrm{\prime }}$. Solving for the middle term and applying [F3] yields ${\int }_s^{t}P{Z}^{\mathrm{\prime }}Q=G(t)-G(s)-{\int }_s^t[P^{\mathrm{\prime }}ZQ+PZ{Q}^{\mathrm{\prime }}]dr$. Thus the remainder after the boundary term differentiates only the two outer propagators; it does not differentiate the whole product $PZQ$ a second time or retain $P{Z}^{\mathrm{\prime }}Q$. By [F1], $P^{\mathrm{\prime }}ZQ+PZ{Q}^{\mathrm{\prime }}=(i\mathrm{/}\mathrm{\hslash })P(HZ-ZA)Q$. Combining this with step 1.1 and $Z^{\mathrm{\prime }}=V$ gives the exact identity $U(t,s)-U_{\mathrm{a}\mathrm{v}\mathrm{g}}(t,s)=-(i\mathrm{/}\mathrm{\hslash })[Z(t)U_{\mathrm{a}\mathrm{v}\mathrm{g}}(t,s)-U(t,s)Z(s)]-{\mathrm{\hslash }}^{-2}{\int }_s^{t}U(t,r)[H(r)Z(r)-Z(r)A]U_{\mathrm{a}\mathrm{v}\mathrm{g}}(r,s)dr$. Every order and sign follows from the displayed three-term product rule.

[Full finite-time constant] Unitarity gives norm at most one to both outer propagators. The two boundary summands in step 2.1 have total norm at most $(\mathrm{\parallel }Z(t)\mathrm{\parallel }+\mathrm{\parallel }Z(s)\mathrm{\parallel })\mathrm{/}\mathrm{\hslash }\le 4b\mathrm{/}(\mathrm{\hslash }\mathrm{\mid }\omega \mathrm{\mid })$. Its remainder integrand has norm at most $(\mathrm{\parallel }H(r)\mathrm{\parallel }+\mathrm{\parallel }A\mathrm{\parallel })\mathrm{\parallel }Z(r)\mathrm{\parallel }\le (2a+2b)2b\mathrm{/}\mathrm{\mid }\omega \mathrm{\mid }$. Applying [F3] and integrating over length $\tau$ gives $\mathrm{\parallel }U(t,s)-U_{\mathrm{a}\mathrm{v}\mathrm{g}}(t,s)\mathrm{\parallel }\le 4b\mathrm{/}(\mathrm{\hslash }\mathrm{\mid }\omega \mathrm{\mid })+4b\tau (a+b)\mathrm{/}({\mathrm{\hslash }}^2\mathrm{\mid }\omega \mathrm{\mid })$. Since $a+b\le a+2b$, this is at most $[4b\mathrm{/}(\mathrm{\hslash }\mathrm{\mid }\omega \mathrm{\mid })][1+\tau (a+2b)\mathrm{/}\mathrm{\hslash }]$, precisely the full restored constant in the Statement. The slightly stronger intermediate estimate is used only to prove that asserted bound.

[Normalized case and all boundary conditions] Taking $\mathrm{\hslash }=1$ in step 3.1 gives the first displayed inequality of the Statement; arbitrary $\mathrm{\hslash }>0$ gives its restored formula without changing $\omega$ or introducing an extra factor into $Z^{\mathrm{\prime }}=V$. The $s=t$ case has $U(s,s)=U_{\mathrm{a}\mathrm{v}\mathrm{g}}(s,s)=I_H$, so the left side is zero and the displayed inequality holds. If $B=0$, both generators coincide and uniqueness gives equality of propagators, again agreeing with the bound. Negative $\omega$ is handled by $\mathrm{\mid }\omega \mathrm{\mid }$ in [F2], and negative absolute times cause no change because [F1] applies on any compact real interval. The zero Hilbert space is included by the norm-at-most-one estimates. CC is used through the actual arbitrary-H Dyson and FTC suppliers; no extra separability, unbounded domain, spectral or AC assumption is introduced. This proves exactly the stated finite-time estimate and retains its explicit absence of a uniform-time claim.