Inverse problem for one-dimensional wave equation with matrix potential

Abstract

We prove a global uniqueness theorem of reconstruction of a matrix-potential a (x, t) {a(x,t)} of one-dimensional wave equation □ u + a u = 0 {\square u+au=0}, x > 0, t > 0 {x>0,t>0}, □ = t 2 - x 2 {\square=\partial-{t}^{2}-\partial-{x}^{2}} with zero Cauchy data for t = 0 {t=0} and given Cauchy data for x = 0 {x=0}, u (0, t) = 0 {u(0,t)=0}, u x (0, t) = g (t) {u-{x}(0,t)=g(t)}. Here u, a, f {u,a,f}, and g are n × n {n\times n} smooth real matrices, det (f (0)) 0 {\det(f(0))\neq 0}, and the matrix t a {\partial-{t}a} is known.