Inverse problem for one-dimensional dynamical Dirac system (BC-method)
Mikhail Belishev, Victor Mikhailov
TL;DR
The paper develops a time-domain inverse problem framework for a one-dimensional dynamical Dirac system using boundary-control methods. It shows that the inverse data, encoded in the response function $r$ on $[0,2T]$, determine the potential $V$ on the triangle $ riangle^T$ if and only if the connecting operator $C^T$, constructed from $r$, is a positive definite isomorphism; it then provides a concrete procedure to recover $V$ from $r$ by solving linear integral equations and extracting $p$ and $q$ from the diagonal kernel values. The approach remains entirely in the time domain, leveraging finite domain of influence and controllability via a composite system $eta^T$, and yields a local, stable reconstruction akin to Gel’fand-Levitan–Krein methods. These contributions advance the BC-method for dynamical IPs and enable direct, data-driven reconstruction of Dirac-type potentials from boundary measurements with potential applications in optics and quantum dynamics.
Abstract
A forward problem for the Dirac system is to find $u=\begin{pmatrix}u_1(x,t)\\u_2(x,t)\end{pmatrix}$ obeying $iu_t+\begin{pmatrix}0&1\\-1&0\end{pmatrix}u_x+\begin{pmatrix}p&q\\q&-p\end{pmatrix}u=0$ for $x>0,\,t>0$;\,\,$u(x,0)=\begin{pmatrix}0\\0\end{pmatrix}$ for $x {\geqslant} 0 $, and $u_1(0,t)=f(t)$ for $t>0$, with the real $p=p(x), q=q(x)$. An input--output map $R: u_1(0,\cdot)\mapsto u_2(0,\cdot)$ is of the convolution form $Rf=if+r\ast f$, where $r=r(t)$ is a {\it response function}. By hyperbolicity of the system, for any $T>0$, function $r\big|_{0 {\leqslant} t {\leqslant} 2T}$ is determined by $p,q\big|_{0 {\leqslant} x {\leqslant} T}$. An inverse problem is: for an (arbitrary) fixed $T>0$, given $r\big|_{0 {\leqslant} t {\leqslant} 2T}$ to recover $p,q\big|_{0 {\leqslant} x {\leqslant} T}$. The procedure that determines $p,q$ is proposed, and the characteristic solvability conditions on $r$ are provided. Our approach is purely time-domain and is based on studying the controllability properties of the Dirac system. In itself the system is not controllable: the local completeness of states does not hold, but its relevant extension gains controllability. It is the fact, which enables one to apply the boundary control method for solving the inverse problem.