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

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