Perturbations on the checked invariant sector

No finite global norm is assigned to creation operators. Nor is graph-domain invariance for arbitrary bounded time-dependent perturbations silently assumed.

For an identical-label regular coupling change $g\mapsto \tilde{g}$, the sector difference has norm ${\Vert \tilde{g}-g\Vert }_2$, giving

Its full autonomous Hamiltonian is separately defined by Theorem thm:C2 when the regularity conditions hold.

This is not the classical input-stability estimate of Em06-B and not an imported Volterra remainder CheungEm06B ({thm:input-stability}). The domain caution also separates this argument from the switched construction in Em03-F CheungEm03F.

Finite-time closeness does not establish perturbed spectral decay at arbitrarily late times.