Lemma: Vacuum Dyson and occupation-tail estimates

For an initial vector consisting of an arbitrary normalized source–carrier state and the reservoir vacuum, the interaction picture Dyson term of order $\ell$ has norm at most

The expansion converges in norm, uniformly on bounded time intervals and uniformly over normalized code-reference inputs.

For the complete prepared isometry,

where $P_m$ acts on the reservoir before regrouping.

Proof. On sectors containing at most $k$ particles,

An $\ell$-fold product acting on vacuum is therefore bounded by $(2G{)}^{\ell }\sqrt{\ell !}$. The ordered integration simplex has volume $t^{\ell }/\ell !$, giving the claimed term bound.

The ratio of consecutive scalar bounds is $2Gt/\sqrt{\ell +1}$, so the series converges. The same proof holds with an untouched reference and carrier. Each term is supported in sectors with at most $\ell$ particles.

For completeness, multiplication of the scalar term bound by any fixed power of $\ell$ still gives a convergent series. Thus the Dyson series converges in every fixed particle-number graph norm. With the finite frequency band and bounded source operators, the integral equation and its derivative hold on the prepared vectors. The resulting Schrödinger-picture solution is the unique solution generated by the self-adjoint Hamiltonian of Theorem thm:selfadjoint. This identifies the expansion with the actual evolution.

Terms of order $\ell \le m$ vanish under $I-P_m$. Summing the remaining bounds proves (eq:occupation-tail).