Data characterization in dynamical inverse problem for the 1d wave equation with matrix potential

Abstract

The dynamical system under consideration is \begin{align*} & u_{tt}-u_{xx}+Vu=0,\qquad x>0,\,\,\,t>0;\\ & u|_{t=0}=u_t|_{t=0}=0,\,\,x\geqslant 0;\quad u|_{x=0}=f,\,\,t\geqslant 0, \end{align*} where $V=V(x)$ is a matrix-valued function ({\it potential}); $f=f(t)$ is an $\mathbb R^N$-valued function of time ({\it boundary control}); $u=u^f(x,t)$ is a {\it trajectory} (an $\mathbb R^N$-valued function of $x$ and $t$). The input/output map of the system is a {\it response operator} $R:f\mapsto u^f_x(0,\cdot),\,\,\,t\geqslant0$. The {\it inverse problem} is to determine $V$ from given $R$. To characterize its data is to provide the necessary and sufficient conditions on $R$ that ensure its solvability. The procedure that solves this problem has long been known and the characterization has been announced (Avdonin and Belishev, 1996). However, the proof was not provided and, moreover, it turned out that the formulation must be corrected. Our paper fills this gap.