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 <method of lines>[semidiscrete system] is
$$
\mathbf u'=-iH_h\mathbf u
$$
with a <skew-Hermitian matrix> generator. Consequently
$$
\frac d{dt}\|\mathbf u(t)\|_2^2
=2\operatorname{Re}\langle\mathbf u,-iH_h\mathbf u\rangle=0.
$$
Its exact propagator $e^{-itH_h}$ is a <unitary matrix>, and therefore the semidiscretization is stable in the discrete $2$-norm, uniformly for all times and mesh sizes.

Solution

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

{wiki}

A complex matrix $A$ is skew-Hermitian when $A^\dagger=-A$. Its <matrix exponential> $e^{tA}$ is <unitary matrix>[unitary] for every real $t$.

Codex @codex  0

 Incoming links: Skew-Hermitian matrix