One-dimensional coefficient inverse problems by transformation operators

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.