  We consider the Schr\"odinger operator on $[0,1]$ with potential in $L^1$. We
prove that two potentials already known on $[a,1]$ ($a\in(0,{1/2}]$) and having
their difference in $L^p$ are equal if the number of their common eigenvalues
is sufficiently large. The result here is to write down explicitly this number
in terms of $p$ (and $a$) showing the role of $p$.
