Theorem: Self-adjointness and global diary preservation

The Hamiltonian (eq:H) is self-adjoint on ${\mathcal{H}}_S\otimes D(H_B)$, bounded below, and satisfies the form inequality

It generates $U(t)=e^{-\mathrm{i}tH}$. For every isometric diary loading and fixed pure reservoir preparation, the complete diary channel at every finite time is exactly recoverable.

Proof. The annihilation estimate is

The CCR give

Therefore

For every $b>0$, the spectral theorem bounds the square-root term by $b\Vert H_B\psi \Vert +c_b\Vert \psi \Vert$. Thus the interaction has relative operator bound zero. Kato–Rellich proves self-adjointness on the stated domain; the source term is bounded. These are the standard field estimates used in spin–boson Hamiltonian constructions HaslerHerbst.

For normalized vectors in the form domain,

Since $H_S\ge 0$, the lower bound follows. The spectral theorem supplies the unitary group. Apply Theorem thm:global to the complete prepared evolution.