  We prove the uniqueness for an inverse problem of determining a matrix
coefficient $P(x)$ of a system of evolution equations $\sigma \ppp_t u =
\ppp_x^2 u(t,x) - P(x) u(t,x)$ for $0<x<\ell$ and $0<t<T$, where $\ell>0$ and
$T>0$ are arbitrarily given. The uniqueness results assert that two solutions
have the same Cauchy data at $x=0$ over $(0,T)$ and the same initial value or
the final value which is positive on $[0,\ell]$, then the zeroth-order
coefficient is uniquely determined on $[0,\ell]$. The uniqueness for inverse
coefficient problem for a system of evolution equations without boundary
conditions over the whole boundary is an open problem even in the one-dimension
in the case where only initial value is given as spatial data. Moreover, in the
case of the zero initial condition, we prove the uniqueness in the half of the
spatial interval.
