Inverse coefficient problem for one-dimensional evolution equation vanishing initial condition

We consider an inverse problem of determining a coefficient $p(x)$ of an evolution equation $\sigma\ppp_tu = a(x)\ppp_x^2u - p(x)u$ for $0<x<\ell$ and $0<t<T$, where $\sigma \in \C \setminus \{0\}$, $\ell>0$ and $T>0$ are arbitrarily given. Our main result is the uniqueness: by assuming that the zeros of initial value $b(x):= u(0,x)$ on $[0, \ell]$ is a finite set and each zero is of order one at most, if two solutions have the same Cauchy data at $x=0$ over $(0,T)$ and the same initial value $b(x)$, then the coefficient $p(x)$ is uniquely determined on $[0,\ell]$.