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 $\sigma {\text{ppp}}_{t}u=a(x){\text{ppp}}_x^{2}u-p(x)u$ for $0<x<\mathrm{\ell }$ and $0<t<T$, where $\sigma \in \text{C}\setminus \{0\}$, $\mathrm{\ell }>0$ 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 $[0,\mathrm{\ell }]$ 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 $[0,\mathrm{\ell }]$.