Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 341 / Section A / 3

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 $u_t=-iHu$,because the inner product $⟨u,Hu⟩$ is real. Thus the Schrodinger equation preserves the $L^2$ norm.

Let $H_h$ be the periodic centered second-difference matrix minus the real diagonal matrix containing $V(x_m)$. It is a Hermitian matrix, so the semidiscrete system iswith a skew-Hermitian matrix generator. ConsequentlyIts exact propagator $e^{-it{H}_h}$ is a unitary matrix, and therefore the semidiscretization is stable in the discrete $2$-norm, uniformly for all times and mesh sizes.

Apply the implicit midpoint rule, equivalently the Crank--Nicolson method, to the semidiscrete equation:It has order two. Its amplification matrix is the Cayley transformBecause $H_h$ is Hermitian, $Q$ is unitary. Hence $\Vert {\mathbf{u}}^{n+1}{\Vert }_2=\Vert {\mathbf{u}}^n{\Vert }_2$ exactly.