Proposition: Energy and bandwidth errors

Compression of field energy to states with particles only in $E$ is $H_E$, with selected Gaussian expectation $\mathrm{tr}(HK)$.

Suppose packets lie in pairwise disjoint subbands whose halfwidths are at most $\Delta$. Several packets may belong to the same subband. Let $H_0$ assign its midpoint to each such packet. Then

Assigning the flag zero energy omits selected energy between zero and ${\omega }_{+}{m}_N(K)$.

Proof. On each $n$-particle sector, energy is the sum of $n$ one-particle operators; compression gives $H$. Using $C_{ij}=K_{ji}$ gives $\mathrm{tr}(HK)$. Multiplication by frequency minus the subband midpoint has norm at most $\Delta$; compression and the $n$-fold sum give the first two estimates. Finally $0⪯H_E⪯{\omega }_{+}{N}_{\mathrm{tot}}$, and both commute with the cutoff projector.