Inverse problem for one-dimensional wave equation with matrix potential

Abstract

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.