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

Abstract

<p style='text-indent:20px;'>The one-dimensional Dirac dynamical system <inline-formula><tex-math id="M1">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> is</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix} $\end{document}</tex-math></inline-formula> is the Pauli matrix; <inline-formula><tex-math id="M3">\begin{document}$ V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ p = p(x) $\end{document}</tex-math></inline-formula> is a potential; <inline-formula><tex-math id="M5">\begin{document}$ u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix} $\end{document}</tex-math></inline-formula> is the trajectory in <inline-formula><tex-math id="M6">\begin{document}$ \mathscr H = L_2(\mathbb R_+;\mathbb C^2) $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M7">\begin{document}$ f\in\mathscr F = L_2([0, \infty);\mathbb C) $\end{document}</tex-math></inline-formula> is a boundary control. System <inline-formula><tex-math id="M8">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> is not controllable: the total reachable set <inline-formula><tex-math id="M9">\begin{document}$ \mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $\end{document}</tex-math></inline-formula> is not dense in <inline-formula><tex-math id="M10">\begin{document}$ \mathscr H $\end{document}</tex-math></inline-formula>, but contains a controllable part <inline-formula><tex-math id="M11">\begin{document}$ \Sigma_u $\end{document}</tex-math></inline-formula>. We construct a dynamical system <inline-formula><tex-math id="M12">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula>, which is controllable in <inline-formula><tex-math id="M13">\begin{document}$ L_2(\mathbb R_+;\mathbb C) $\end{document}</tex-math></inline-formula> and connected with <inline-formula><tex-math id="M14">\begin{document}$ \Sigma_u $\end{document}</tex-math></inline-formula> via a unitary transform. The construction is based on geometrical optics relations: trajectories of <inline-formula><tex-math id="M15">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula> are composed of jump amplitudes that arise as a result of projecting in <inline-formula><tex-math id="M16">\begin{document}$ \overline{\mathscr U} $\end{document}</tex-math></inline-formula> onto the reachable sets <inline-formula><tex-math id="M17">\begin{document}$ \mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $\end{document}</tex-math></inline-formula>. System <inline-formula><tex-math id="M18">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula>, which we call the <i>amplitude model</i> of the original <inline-formula><tex-math id="M19">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>, has the same input/output correspondence as system <inline-formula><tex-math id="M20">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>. As such, <inline-formula><tex-math id="M21">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula> provides a canonical completely reachable realization of the Dirac system.</p>