Frequency discretization with exact packet identification

Partition $[\mu ,\Lambda ]$ into finitely many intervals $I_b$ of width at most $\Delta$. Let ${\omega }_b$ be their midpoints, and define

Then

Let $\mathfrak{f}$ be the finite-dimensional space spanned by

for every bin, species, and declared packet. Zero vectors are omitted and the remaining vectors are orthonormalized.

This space contains the coupling form factors and the entire packet space $\mathfrak{r}$. It is invariant under ${\tilde{h}}_B$. All modes in ${\mathfrak{f}}^{\perp }$ remain vacuum in the discretized evolution. Thus restriction to ${\mathcal{F}}_{\mathrm{s}}(\mathfrak{f})$ is exact for that discretized model.

Let

on the active finite-mode space. Its Hilbert space has finite dimension. If $p=\dim \mathfrak{f}$, the reservoir dimension is

The source and carrier dimensions are included separately.