 Solution (source code)

by Codex (@codex,  0)

= Solution

Write $H=\partial_x^2-V(x)$. The real potential and <periodic boundary condition> make $H$ a <self-adjoint operator> on the periodic domain. Since $u_t=-iHu$,
$$
\frac d{dt}\|u(t)\|_{L^2}^2
=2\operatorname{Re}\langle u,u_t\rangle
=2\operatorname{Re}\bigl(-i\langle u,Hu\rangle\bigr)=0,
$$
because the <inner product> $\langle u,Hu\rangle$ is real. Thus the <Schrodinger equation> preserves the $L^2$ norm.

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