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

TL;DR

This paper addresses the inverse problem of determining a matrix-valued potential in a 1D wave equation from boundary response data, providing a rigorous proof and correction of the existing characterization conditions.

Contribution

It offers a complete proof and correction of the data characterization for the inverse problem of the 1D wave equation with matrix potential.

Findings

Provided a rigorous proof of the inverse problem characterization.

Corrected the previously announced data characterization formulation.

Established necessary and sufficient conditions for the response operator.

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.