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

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The first author is supported by the RFBR grant 20-01 627A and Volkswagen Foundation. The second author is supported by the RFBR grant 19-01-00565A

• 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.

Mathematics Subject Classification: Primary: 35B30, 47N70, 46N20; Secondary: 93B28.

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