Theorem: Exact microscopic conservation and bounded reservoir energy

For every logical input, define

Then

Uniformly in the logical input and time,

The source current, with positive sign for loss from the source, is

Consequently,

No sign is asserted for either current at every time.

Proof. The initially prepared vectors belong to $D(H)$. Autonomous unitary evolution preserves the total Hamiltonian expectation. The initial bath and interaction means vanish, giving (eq:energy-conservation).

The form inequality (eq:lower-H) gives

which proves (eq:bath-energy-bound).

The particle-sector bounds established in Section sec:regulator also show that these prepared evolved vectors have finite moments of every fixed particle-number order on bounded time intervals. Since $\omega$ is bounded on the one-particle band, the commutator expectations can be obtained by finite-particle approximation. Differentiation gives ${\dot{E}}_S=\mathrm{i}⟨[H,H_S]⟩=\mathrm{i}⟨[V,H_S]⟩$. Differentiating (eq:energy-conservation) proves (eq:interaction-energy).