A canonical model of the one-dimensional dynamical Dirac system with boundary control

The one-dimensional Dirac dynamical system $ \Sigma $ is

$ \begin{align*} & iu_t+i\sigma_{\!_3}\, u_x+Vu = 0, \, \, \, \, x, t>0;\, \, \, u|_{t = 0} = 0, \, \, x>0;\, \, \, \, u_1|_{x = 0} = f, \, \, t>0, \end{align*} $

where $ \sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix} $ is the Pauli matrix; $ V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix} $ with $ p = p(x) $ is a potential; $ u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix} $ is the trajectory in $ \mathscr H = L_2(\mathbb R_+;\mathbb C^2) $; $ f\in\mathscr F = L_2([0, \infty);\mathbb C) $ is a boundary control. System $ \Sigma $ is not controllable: the total reachable set $ \mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $ is not dense in $ \mathscr H $, but contains a controllable part $ \Sigma_u $. We construct a dynamical system $ \Sigma_a $, which is controllable in $ L_2(\mathbb R_+;\mathbb C) $ and connected with $ \Sigma_u $ via a unitary transform. The construction is based on geometrical optics relations: trajectories of $ \Sigma_a $ are composed of jump amplitudes that arise as a result of projecting in $ \overline{\mathscr U} $ onto the reachable sets $ \mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $. System $ \Sigma_a $, which we call the amplitude model of the original $ \Sigma $, has the same input/output correspondence as system $ \Sigma $. As such, $ \Sigma_a $ provides a canonical completely reachable realization of the Dirac system.