Reference-preserving regulator control

Let $h_L$ be the restriction of the one-particle Hamiltonian to sites $0,\dots ,L$, with the bond $L,L+1$ removed. Embed its evolution into the infinite one-particle space. Unused exterior modes start in vacuum.

For $L\ge M$, let ${{\mathcal{N}}_{t,\mathcal{L}}}^{(M)}$ denote its channel to the same collected region $R_M$.