Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 5 / a / Solution

Write $H=-{\partial }_x^2+V(x)$. The real potential and periodic boundary conditions make $H$ a self-adjoint operator. Since $u_t=-iHu$,because the expectation of a self-adjoint operator is real. Equivalently, integrating the kinetic term by parts leaves $\int |u_x{|}^{2}dx$ and the periodic boundary term cancels. Thus the continuous Schrodinger equation preserves its $L^2$ norm.