Proposition: Well-posed evolution

For bounded piecewise continuous real $v,\kappa$, $K(u)$ has a unique interaction-picture unitary propagator.

Proof. Relative to $h\oplus 1$, the perturbation is bounded, self-adjoint, and finite rank. Strong continuity on the finitely many packet vectors gives norm continuity of the corresponding rank-one interaction-picture operators on each control segment. The norm-convergent Dyson series gives existence and uniqueness; self-adjointness gives unitarity.