= Solution
Write $H=-\partial_x^2+V(x)$. The real potential and periodic boundary conditions make $H$ a <self-adjoint operator>. Since $u_t=-iHu$,
$$
\frac d{dt}\int_{-1}^1|u|^2dx
=2\operatorname{Re}\int_{-1}^1\overline u,u_tdx
=2\operatorname{Re}\left(-i\int_{-1}^1\overline u,Hu\,dx\right)=0,
$$
because the expectation of a self-adjoint operator is real. Equivalently, integrating the kinetic term by parts leaves $\int|u_x|^2dx$ and the periodic boundary term cancels. Thus the continuous <Schrodinger equation> preserves its $L^2$ norm.
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