= Norm conservation of a semidiscrete Schrödinger equation
{title2=$iU'=HU,\ H=H^*\ \Longrightarrow\ \|U(t)\|=\|U(0)\|$}
If a spatial <method of lines> has a <Hermitian matrix> $H$, its generator $-iH$ is a <skew-Hermitian matrix>. The <matrix exponential> $e^{-itH}$ is a <unitary matrix>, so it preserves the <Euclidean norm> and any constant-volume <discrete L2 norm>. Equivalently, differentiating the squared <norm> gives $2\operatorname{Re}(-iU^*HU)=0$. A real sampled potential preserves the Hermitian property of a symmetric discrete Laplacian. Subsequent time discretization must be assessed separately.
Back to article page
