A scattering endpoint derived from the Hamiltonian

The unmodified Schrödinger-picture reservoir state need not converge: its wavepacket continues moving. An endpoint must therefore specify an asymptotic outgoing representation.

Let

and let $h_B$ be the same uniform half-line Hamiltonian on sites $j\ge 1$, with the source bond absent. On the full one-particle space set

The difference $h-h_{\mathrm{dec}}$ has finite rank.

scattering input. For self-adjoint operators differing by a trace-class operator, the wave operators on their absolutely continuous subspaces exist and are complete Kato. Only this theorem is imported. The required spectral and finite-rank hypotheses for the present model are checked below.